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104  <h3 id="/publications/2025-fwsafetyradius">Tropical Fermat-Weber Points over Bergman Fans</h3>
105  <div class="authors">
106    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
107    John Sabol, 
108    Roan Talbut, 
109    Ruriko Yoshida
110    
111  </div>
112  
113    <div class="venue">
114      
115       <i>accepted</i> 
116      Vietnam Journal of Mathematics
117      
118      2025
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129    <div class="extra-links">
130    
131      <a href="https://arxiv.org/abs/2505.09584">
132        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
133      </a>
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149<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2025-maxout.html&quot;,&quot;relative_path&quot;:&quot;_publications/2025-maxout.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2025-fwsafetyradius.html&quot;,&quot;relative_path&quot;:&quot;_publications/2025-fwsafetyradius.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2024-tropbg.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-tropbg.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2024-homaloidal.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-homaloidal.html&quot;,&quot;id&quot;:&quot;/publications/2024-homaloidal&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2024-homaloidal&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2024,&quot;title&quot;:&quot;Homaloidal polynomials and Gaussian models of maximum likelihood degree 1&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Pratik Misra&quot;,&quot;Pardis Semnani&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Algebraic Statistics&quot;,&quot;pdf&quot;:&quot;https://msp.org/astat/2024/15-2/p02.xhtml&quot;,&quot;tags&quot;:[&quot;ML degree&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2024-homaloidal&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2024-tropbg&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2025-fwsafetyradius.html&quot;,&quot;relative_path&quot;:&quot;_publications/2025-fwsafetyradius.html&quot;,&quot;id&quot;:&quot;/publications/2025-fwsafetyradius&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2025-fwsafetyradius&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2025,&quot;title&quot;:&quot;Tropical Fermat-Weber Points over Bergman Fans&quot;,&quot;status&quot;:&quot;accepted&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;John Sabol&quot;,&quot;Roan Talbut&quot;,&quot;Ruriko Yoshida&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;venue_tags&quot;:null,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2505.09584&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;highlight&quot;:true,&quot;slug&quot;:&quot;2025-fwsafetyradius&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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150  <h3 id="/publications/2025-maxout">Maxout polytopes</h3>
151  <div class="authors">
152    Andrei Balakin, 
153    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
154    Georg Loho, 
155    Bernd Sturmfels
156    
157  </div>
158  
159    <div class="venue">
160      
161       <i>preprint</i> 
162      arXiv
163      
164      2025
165    </div>
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175    <div class="extra-links">
176    
177      <a href="https://arxiv.org/abs/2509.21286">
178        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
179      </a>
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191    </div>
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193</div>
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195<h2 id="y2024" class="year">2024</h2>
196
197<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;
197The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;
197\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2024-3sunlet&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-bmtmMLD.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-bmtmMLD.html&quot;,&quot;id&quot;:&quot;/publications/2024-bmtmMLD&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2024-bmtmMLD&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2024,&quot;title&quot;:&quot;Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance&quot;,&quot;status&quot;:&quot;accepted&quot;,&quot;authors&quot;:[&quot;Jane Coons&quot;,&quot;Shelby Cox&quot;,&quot;Aida Maraj&quot;,&quot;Ikenna Nometa&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;
197Journal of Symbolic Computation, MEGA 2024 Special Edition&quot;,&quot;venue_tags&quot;:&quot;JSC&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2402.10322&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;ML degree&quot;],&quot;highlight&quot;:true,&quot;slug&quot;:&quot;2024-bmtmMLD&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; 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197 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2024-3sunlet&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2024,&quot;title&quot;:&quot;Group-Based Phylogenetic Models on 3-Sunlet Networks&quot;,&quot;status&quot;:&quot;Open access&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Elizabeth Gross&quot;,&quot;Samuel Martin&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Bulletin of Mathematical Biology&quot;,&quot;venue_tags&quot;:&quot;BMB&quot;,&quot;pdf&quot;:&quot;https://rdcu.be/eA9b3&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:true,&quot;slug&quot;:&quot;2024-3sunlet&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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198  <h3 id="/publications/2024-3sunlet">Group-Based Phylogenetic Models on 3-Sunlet Networks</h3>
199  <div class="authors">
200    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
201    Elizabeth Gross, 
202    Samuel Martin
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208       <i>Open access</i> 
209      Bulletin of Mathematical Biology
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211      2024
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224      <a href="https://rdcu.be/eA9b3">
225        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
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243  <h3 id="/publications/2024-bmtmMLD">Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance</h3>
244  <div class="authors">
245    Jane Coons, 
246    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
247    Aida Maraj, 
248    Ikenna Nometa
249    
250  </div>
251  
252    <div class="venue">
253      
254       <i>accepted</i> 
255      Journal of Symbolic Computation, MEGA 2024 Special Edition
256      
257      2024
258    </div>
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268    <div class="extra-links">
269    
270      <a href="https://arxiv.org/abs/2402.10322">
271        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
272      </a>
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284    </div>
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289  <h3 id="/publications/2024-homaloidal">Homaloidal polynomials and Gaussian models of maximum likelihood degree 1</h3>
290  <div class="authors">
291    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
292    Pratik Misra, 
293    Pardis Semnani
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295  </div>
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300      Algebraic Statistics
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302      2024
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316        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
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334  <h3 id="/publications/2024-tropbg">Tropicalizing binary geometries</h3>
335  <div class="authors">
336    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
337    Igor Makhlin
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343       <i>published</i> 
344      Le Matematiche
345      
346      2024
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359      <a href="https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/2744">
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377<h2 id="y2023" class="year">2023</h2>
378
379<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;excerpt&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;previous&quot;:null,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
379ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot;,\&quot;headline\&quot;:\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
379ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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379 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Tree Topologies along a Tropical Line Segment | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Ruriko Yoshida\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
379o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree space
379s is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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379 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). 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In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;
379The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length;
379 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). 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380  <h3 id="/publications/2023+-weightedfw">The tropical polytope is the set of all weighted tropical Fermat-Weber points</h3>
381  <div class="authors">
382    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
383    Mark Curiel
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389       <i>preprint</i> 
390      arXiv
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392      2023
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405      <a href="https://arxiv.org/abs/2310.07732.pdf">
406        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
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423<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
423o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Tree Topologies along a Tropical Line Segment | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Ruriko Yoshida\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
423o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree space
423s because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;
423The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;
423The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;
423\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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423 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;
423The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;
423\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2024-3sunlet&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-bmtmMLD.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-bmtmMLD.html&quot;,&quot;id&quot;:&quot;/publications/2024-bmtmMLD&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2024-bmtmMLD&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2024,&quot;title&quot;:&quot;Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance&quot;,&quot;status&quot;:&quot;accepted&quot;,&quot;authors&quot;:[&quot;Jane Coons&quot;,&quot;Shelby Cox&quot;,&quot;Aida Maraj&quot;,&quot;Ikenna Nometa&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;
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423 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). 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In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;
423The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length;
423 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;}">
424  <h3 id="/publications/2023-classtop">Classifying Tree Topology Changes along Tropical Line Segments</h3>
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426    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>
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466<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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466 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). 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In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;
466The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length;
466 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;relative_path&quot;:&quot;_publications/2024-3sunlet.html&quot;,&quot;id&quot;:&quot;/publications/2024-3sunlet&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2024-3sunlet&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2024,&quot;title&quot;:&quot;Group-Based Phylogenetic Models on 3-Sunlet Networks&quot;,&quot;status&quot;:&quot;Open access&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Elizabeth Gross&quot;,&quot;Samuel Martin&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Bulletin of Mathematical Biology&quot;,&quot;venue_tags&quot;:&quot;BMB&quot;,&quot;pdf&quot;:&quot;https://rdcu.be/eA9b3&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:true,&quot;slug&quot;:&quot;2024-3sunlet&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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466 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;}">
467  <h3 id="/publications/2023-regmixup">[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification</h3>
468  <div class="authors">
469    Dylan Cordaro, 
470    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
471    Yiman Ren, 
472    Teresa Yu
473    
474  </div>
475  
476    <div class="venue">
477      
478        <a href="https://paperswithcode.com/rc2022">
479      
480      
481      2022 Reproducibility Challenge
482      </a>
483      2023
484    </div>
485  
486
487  
488
489  
490
491  
492
493  
494    <div class="extra-links">
495    
496      <a href="https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf">
497        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
498      </a>
499    
500    
501    
502    
503    
504    
505    
506    
507    
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509    
510    </div>
511  
512</div>
513
514<h2 id="y2022" class="year">2022</h2>
515
516<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;excerpt&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;previous&quot;:null,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree space
516s because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Tree Topologies along a Tropical Line Segment | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Ruriko Yoshida\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree space
516s because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree space
516s because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
516ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot;,\&quot;headline\&quot;:\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
516ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
516o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Tree Topologies along a Tropical Line Segment | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Ruriko Yoshida\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
516o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree space
516s because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;excerpt&quot;:&quot;
516The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-regmixup.html&quot;,&quot;id&quot;:&quot;/publications/2023-regmixup&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-regmixup&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification&quot;,&quot;authors&quot;:[&quot;Dylan Cordaro&quot;,&quot;Shelby Cox&quot;,&quot;Yiman Ren&quot;,&quot;Teresa Yu&quot;],&quot;type&quot;:&quot;reproducibility&quot;,&quot;venue&quot;:&quot;2022 Reproducibility Challenge&quot;,&quot;pdf&quot;:&quot;https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf&quot;,&quot;venue_url&quot;:&quot;https://paperswithcode.com/rc2022&quot;,&quot;tags&quot;:[&quot;machine learning&quot;,&quot;reproducibility&quot;],&quot;slug&quot;:&quot;2023-regmixup&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;
516The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\&quot;,\&quot;headline\&quot;:\&quot;Classifying Tree Topology Changes along Tropical Line Segments\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2023-classtop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;
516\n  The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The space of phylogenetic trees arises naturally in tropical geometry as the tropical Grassmannian. Tropical geometry therefore suggests a natural notion of a tropical path between two trees, given by a tropical line segment in the tropical Grassmannian. It was previously conjectured that tree topologies along such a segment change by a combinatorial operation known as Nearest Neighbor Interchange (NNI). We provide counterexamples to this conjecture, but prove that changes in tree topologies along the tropical line segment are either NNI moves or “four clade rearrangement” moves for generic trees. In addition, we show that the number of NNI moves occurring along the tropical line segment can be as large as n2, but the average number of moves when the two endpoint trees are chosen at random is O(n(log n)4). This is in contrast with O(n log n), the average number of NNI moves needed to transform one tree into another.\n&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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516 i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;&quot;,&quot;url&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;The tropical polytope is the set of all weighted tropical Fermat-Weber points&quot;,&quot;status&quot;:&quot;preprint&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Mark Curiel&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;arXiv&quot;,&quot;venue_tags&quot;:&quot;arXiv&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/abs/2310.07732.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023+-weightedfw&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Tree Topologies along a Tropical Line Segment | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Ruriko Yoshida\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if tw
516o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree space
516s is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;}">
517  <h3 id="/publications/2022-treetop">Tree Topologies along a Tropical Line Segment</h3>
518  <div class="authors">
519    Ruriko Yoshida, 
520    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>
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522  </div>
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527      Vietnam Journal of Mathematics
528      
529      2022
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542      <a href="https://doi.org/10.1007/s10013-021-00526-3">
543        <i class="far fa-file-pdf" aria-hidden="true"></i> PDF
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553      <a href="https://www.doi2bib.org/bib/10.1007/s10013-021-00526-3">
554        <i class="fas fa-book" aria-hidden="true"></i> Bibtex
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564<h2 id="y2019" class="year">2019</h2>
565
566<div class="publication" data-pub="{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;excerpt&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
566ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;previous&quot;:null,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;excerpt&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;relative_path&quot;:&quot;_publications/2019-eulerchar.html&quot;,&quot;excerpt&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. 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We also explore the method of vertex operator algebras.&quot;,&quot;previous&quot;:null,&quot;id&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
566ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. 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We also explore the method of vertex operator algebras.\&quot;,\&quot;headline\&quot;:\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
566ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]},&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023+-weightedfw.html&quot;,&quot;excerpt&quot;:&quot;&quot;,&quot;previous&quot;:{&quot;path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2022-treetop.html&quot;,&quot;id&quot;:&quot;/publications/2022-treetop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;id&quot;:&quot;/publications/2023+-weightedfw&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;next&quot;:{&quot;path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;relative_path&quot;:&quot;_publications/2023-classtop.html&quot;,&quot;id&quot;:&quot;/publications/2023-classtop&quot;,&quot;collection&quot;:&quot;publications&quot;,&quot;url&quot;:&quot;/publications/2023-classtop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2023,&quot;title&quot;:&quot;Classifying Tree Topology Changes along Tropical Line Segments&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Algebraic Statistics&quot;,&quot;venue_tags&quot;:&quot;Alg Stat&quot;,&quot;pdf&quot;:&quot;https://arxiv.org/pdf/2302.03611.pdf&quot;,&quot;tags&quot;:[&quot;tropical geometry&quot;,&quot;phylogenetics&quot;],&quot;highlight&quot;:false,&quot;slug&quot;:&quot;2023-classtop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; 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566o given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Tree Topologies along a Tropical Line Segment\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Ruriko Yoshida\&quot;},\&quot;description\&quot;:\&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\&quot;,\&quot;headline\&quot;:\&quot;Tree Topologies along a Tropical Line Segment\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2022-treetop\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree space
566s is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.&quot;,&quot;url&quot;:&quot;/publications/2022-treetop&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2022,&quot;title&quot;:&quot;Tree Topologies along a Tropical Line Segment&quot;,&quot;authors&quot;:[&quot;Ruriko Yoshida&quot;,&quot;Shelby Cox&quot;],&quot;doi&quot;:&quot;10.1007/s10013-021-00526-3&quot;,&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Vietnam Journal of Mathematics&quot;,&quot;pdf&quot;:&quot;https://doi.org/10.1007/s10013-021-00526-3&quot;,&quot;tags&quot;:[&quot;phylogenetics&quot;,&quot;tropical geometry&quot;],&quot;slug&quot;:&quot;2022-treetop&quot;,&quot;ext&quot;:&quot;.html&quot;},&quot;output&quot;:&quot;&lt;!DOCTYPE html&gt;\n&lt;html lang=\&quot;en-US\&quot;&gt;\n  &lt;head&gt;\n  &lt;meta charset=\&quot;utf-8\&quot;&gt;\n  &lt;meta http-equiv=\&quot;x-ua-compatible\&quot; content=\&quot;ie=edge\&quot;&gt;\n  &lt;meta name=\&quot;viewport\&quot; content=\&quot;width=device-width, initial-scale=1\&quot;&gt;\n  &lt;meta name=\&quot;application-name\&quot; content=\&quot;Shelby Cox\&quot;&gt;\n  &lt;meta name=\&quot;theme-color\&quot; content=\&quot;#b00\&quot;&gt;\n  &lt;link rel=\&quot;shortcut icon\&quot; href=\&quot;/favicon.ico\&quot;/&gt;\n  &lt;link rel=\&quot;icon\&quot; type=\&quot;image/png\&quot; href=\&quot;/favicon.png\&quot; sizes=\&quot;250x250\&quot; /&gt;\n\n  &lt;!-- Begin Jekyll SEO tag v2.8.0 --&gt;\n&lt;title&gt;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox&lt;/title&gt;\n&lt;meta name=\&quot;generator\&quot; content=\&quot;Jekyll v3.10.0\&quot; /&gt;\n&lt;meta property=\&quot;og:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;meta name=\&quot;author\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:locale\&quot; content=\&quot;en_US\&quot; /&gt;\n&lt;meta name=\&quot;description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
566ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;meta property=\&quot;og:description\&quot; content=\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot; /&gt;\n&lt;link rel=\&quot;canonical\&quot; href=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:url\&quot; content=\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot; /&gt;\n&lt;meta property=\&quot;og:site_name\&quot; content=\&quot;Shelby Cox\&quot; /&gt;\n&lt;meta property=\&quot;og:type\&quot; content=\&quot;website\&quot; /&gt;\n&lt;meta name=\&quot;twitter:card\&quot; content=\&quot;summary\&quot; /&gt;\n&lt;meta property=\&quot;twitter:title\&quot; content=\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot; /&gt;\n&lt;script type=\&quot;application/ld+json\&quot;&gt;\n{\&quot;@context\&quot;:\&quot;https://schema.org\&quot;,\&quot;@type\&quot;:\&quot;WebPage\&quot;,\&quot;author\&quot;:{\&quot;@type\&quot;:\&quot;Person\&quot;,\&quot;name\&quot;:\&quot;Shelby Cox\&quot;},\&quot;description\&quot;:\&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\&quot;,\&quot;headline\&quot;:\&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\&quot;,\&quot;url\&quot;:\&quot;https://www.shelbycox.github.io/publications/2019-eulerchar\&quot;}&lt;/script&gt;\n&lt;!-- End Jekyll SEO tag --&gt;\n\n\n  &lt;link rel=\&quot;alternate\&quot; type=\&quot;application/rss+xml\&quot; title=\&quot;Shelby Cox\&quot; href=\&quot;https://www.shelbycox.github.io/feed.xml\&quot;&gt;\n\n  &lt;link href=\&quot;https://use.fontawesome.com/releases/v6.2.1/css/all.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n  &lt;link href=\&quot;/styles.css\&quot; rel=\&quot;stylesheet\&quot;&gt;\n\n  \n&lt;/head&gt;\n\n  &lt;body&gt;\n    \n&lt;header class=\&quot;page-header\&quot;&gt;\n  &lt;nav class=\&quot;container\&quot;&gt;\n    &lt;a class=\&quot;site-title\&quot; href=\&quot;/\&quot;&gt;Shelby Cox&lt;/a&gt;\n\n    &lt;a href=\&quot;/projects/\&quot; &gt;Projects&lt;/a&gt;\n    &lt;a href=\&quot;/publications/\&quot; aria-current=\&quot;page\&quot;&gt;Papers&lt;/a&gt;\n    &lt;!-- &lt;a href=\&quot;/talks/\&quot; &gt;Talks&lt;/a&gt; --&gt;\n    &lt;!-- &lt;a href=\&quot;/blog/\&quot; &gt;Blog&lt;/a&gt; --&gt;\n    &lt;a href=\&quot;/assets/CV_24112025.pdf\&quot; &gt;CV&lt;/a&gt;\n\n    &lt;span class=\&quot;external\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox\&quot;&gt;&lt;i class=\&quot;fab fa-github\&quot; aria-hidden=\&quot;true\&quot;&gt;&lt;/i&gt; GitHub&lt;/a&gt;\n    &lt;/span&gt;\n  &lt;/nav&gt;\n&lt;/header&gt;\n\n\n    &lt;div class=\&quot;page-content\&quot;&gt;\n      &lt;section class=\&quot;container \&quot;&gt;\n  The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated You
566ng diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.\n&lt;/section&gt;\n\n    &lt;/div&gt;\n\n    &lt;footer&gt;\n  &lt;div class=\&quot;container\&quot;&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      Hello from Ann Arbor.&lt;br/&gt;\n      &lt;abbr title=\&quot;Last build on 2026-01-07\&quot;&gt;January 2026&lt;/abbr&gt;\n    &lt;/div&gt;\n    &lt;div class=\&quot;footer-col\&quot;&gt;\n      &lt;a href=\&quot;https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\&quot;&gt;Edit this page&lt;/a&gt;\n    &lt;/div&gt;\n  &lt;/div&gt;\n&lt;/footer&gt;\n\n    &lt;script&gt;\n  function trim(str) {\n    return str.replace(/^\\s+|\\s+$/g, '');\n  }\n  var headers = document.querySelectorAll(\&quot;h2, h3, h4, h5, h6\&quot;);\n  for (var i=0; i&lt;headers.length; i++) {\n    var h = headers[i];\n    var name = h.getAttribute(\&quot;id\&quot;);\n    var title = h.innerHTML;\n    h.innerHTML = '&lt;a href=\&quot;#' + name + '\&quot; class=\&quot;anchor\&quot;&gt;&lt;i class=\&quot;fas fa-hashtag\&quot;&gt;&lt;/i&gt;&lt;/a&gt;' + trim(title);\n  }\n&lt;/script&gt;\n\n  &lt;/body&gt;\n&lt;/html&gt;\n&quot;,&quot;content&quot;:&quot;The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.&quot;,&quot;url&quot;:&quot;/publications/2019-eulerchar&quot;,&quot;draft&quot;:false,&quot;categories&quot;:[],&quot;layout&quot;:&quot;publication&quot;,&quot;year&quot;:2019,&quot;title&quot;:&quot;Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams&quot;,&quot;authors&quot;:[&quot;Shelby Cox&quot;,&quot;Amal Mattoo&quot;],&quot;type&quot;:[&quot;research&quot;],&quot;venue&quot;:&quot;Minnesota Journal of Undergraduate Mathematics&quot;,&quot;pdf&quot;:&quot;https://pubs.lib.umn.edu/index.php/mjum/article/view/4137&quot;,&quot;slug&quot;:&quot;2019-eulerchar&quot;,&quot;ext&quot;:&quot;.html&quot;,&quot;tags&quot;:[]}">
567  <h3 id="/publications/2019-eulerchar">Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams</h3>
568  <div class="authors">
569    <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a>, 
570    Amal Mattoo
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577      Minnesota Journal of Undergraduate Mathematics
578      
579      2019
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592      <a href="https://pubs.lib.umn.edu/index.php/mjum/article/view/4137">
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613  (function() {
614  var pubElems = document.querySelectorAll(".publication");
615  var yearElems = document.querySelectorAll(".year");
616
617  var clearElem = document.getElementById("clear-filters");
618  var highlightElem = document.getElementById("highlight");
619
620  var data = [];
621  var allYears = new Set();
622
623  pubElems.forEach(function(element) {
624    var item = JSON.parse(element.getAttribute("data-pub"));
625
626    if (item.highlight) {
627      item.highlight = "Yes";
628    }
629
630    allYears.add(item.year);
631
632    item.element = element;
633
634    item.authors = (item.authors || []).filter(function(d) {
635      return d !== "Dominik Moritz";
636    });
637
638    data.push(item);
639  });
640
641  var engine = itemsjs(data, {
642    aggregations: {
643      venue_tags: {
644        size: 5
645      },
646      authors: {
647        size: 6
648      },
649      awards: {
650        size: 5
651      },
652      highlight: {
653        size: 1
654      },
655      tags: {
656        size: 6
657      },
658      type: {
659        size: 5
660      }
661    },
662    searchableFields: ["authors", "awards", "tags", "type", "title", "content"]
663  });
664
665  // get default search from URL
666  var hash = decodeURIComponent(window.location.hash.substr(1));
667
668  var result = hash.split('&').reduce(function (res, item) {
669      var [key, value] = item.split('=');
670      if (key && value) {
671        if (key in res) {
672          res[key].push(value)
673        } else {
674          res[key] = [value];
675        }
676      }
677      return res;
678  }, {});
679
680  var query = { filters: result };
681
682  function setAggs(aggs) {
683    document.querySelectorAll("#facets > .facet").forEach(function(facet) {
684      var id = facet.getAttribute("id");
685
686      var buckets = aggs[id].buckets;
687
688      var el = facet.querySelector("ul");
689      if (buckets.length === 0) {
690        el.innerHTML = "Empty";
691      } else {
692        el.innerHTML = "";
693
694        buckets.forEach(function(bucket) {
695          if (query.filters[id] && query.filters[id].indexOf(bucket.key) >= 0) {
696            bucket.in_query = true;
697          }
698        });
699
700        var maxDocCount = Math.max.apply(
701          null,
702          buckets.map(function(bucket) {
703            return bucket.doc_count;
704          })
705        );
706
707        buckets.forEach(function(bucket) {
708          var child = document.createElement("li");
709
710          var wrap = document.createElement("span");
711          child.appendChild(wrap);
712
713          var text = document.createElement("span");
714          text.classList.add("limited");
715          text.innerText = bucket.key;
716          text.setAttribute("title", bucket.key);
717          var number = document.createElement("span");
718          number.classList.add("cnt");
719          number.innerText = " (" + bucket.doc_count + ")";
720          wrap.appendChild(text);
721          wrap.appendChild(number);
722
723          var barFull = document.createElement("div");
724          barFull.classList.add("bar-full");
725          child.append(barFull);
726
727          var bar = document.createElement("div");
728          bar.classList.add("bar");
729          bar.style.width = "" + (bucket.doc_count / maxDocCount) * 100 + "%";
730          barFull.append(bar);
731
732          if (bucket.in_query) {
733            child.classList.add("in-query");
734
735            // remove filter
736            child.onclick = function() {
737              query.filters[id].splice(
738                query.filters[id].indexOf(bucket.key),
739                1
740              );
741              if (query.filters[id].length === 0) {
742                delete query.filters[id];
743              }
744              search(query);
745            };
746          } else {
747            // add to filter
748            child.onclick = function() {
749              if (query.filters[id]) {
750                query.filters[id].push(bucket.key);
751              } else {
752                query.filters[id] = [bucket.key];
753              }
754              search(query);
755            };
756          }
757
758          el.appendChild(child);
759        });
760      }
761    });
762  }
763
764  var ftSearch = document.getElementById("ft-search");
765  ftSearch.oninput = function() {
766    var val = ftSearch.value;
767
768    if (val) {
769      query.query = val;
770    } else {
771      delete query.query;
772    }
773
774    search(query);
775  }
776
777  function search(query) {
778    console.time("Search");
779
780    var result = engine.search(Object.assign({ per_page: data.length }, query));
781
782    setAggs(result.data.aggregations);
783
784    var counter = pubElems.length - result.data.items.length;
785
786    document.getElementById("count_hidden").innerText = counter;
787    document.getElementById("count_total").innerText = pubElems.length;
788
789    pubElems.forEach(function(element) {
790      element.classList.add("hidden");
791    });
792
793    var visibleYears = {};
794    result.data.items.forEach(function(item) {
795      item.element.classList.remove("hidden");
796      visibleYears[item.year] = 1;
797    });
798
799    yearElems.forEach(function(element) {
800      element.classList.add("hidden");
801    });
802    allYears.forEach(function(year) {
803      if (year in visibleYears) {
804        document.getElementById("y" + year).classList.remove("hidden");
805      }
806    });
807
808    // show or hide notification about filtered papers
809    if (Object.keys(query.filters).length || query.query) {
810      clearElem.classList.remove("hidden");
811    } else {
812      clearElem.classList.add("hidden");
813    }
814
815    highlightElem.checked = !!query.filters.highlight;
816
817    console.timeEnd("Search");
818  }
819
820  highlightElem.onchange = function() {
821    if (highlightElem.checked) {
822      query.filters.highlight = ["Yes"];
823    } else {
824      delete query.filters.highlight;
825    }
826    search(query);
827  };
828
829  clearElem.onclick = function() {
830    query = { filters: {} };
831    ftSearch.value = "";
832    search(query);
833  };
834
835  search(query);
836
837  document.getElementById("facets").classList.remove("hidden");
838  document.getElementById("only-highlight").classList.remove("hidden");
839})();
840
841</script>
841
842
843
844</section>
845
846    </div>
847
848    <footer>
849  <div class="container">
850    <div class="footer-col">
851      Hello from Ann Arbor.<br>
852      <abbr title="Last build on 2026-01-07">January 2026</abbr>
853    </div>
854    <div class="footer-col">
855      <a href="https://github.com/shelbycox/shelbycox.github.io/edit/master/publications.md">Edit this page</a>
856    </div>
857  </div>
858</footer>
859
860    
860<script>
861  function trim(str) {
862    return str.replace(/^\s+|\s+$/g, '');
863  }
864  var headers = document.querySelectorAll("h2, h3, h4, h5, h6");
865  for (var i=0; i<headers.length; i++) {
866    var h = headers[i];
867    var name = h.getAttribute("id");
868    var title = h.innerHTML;
869    h.innerHTML = '<a href="#' + name + '" class="anchor"><i class="fas fa-hashtag"></i></a>' + trim(title);
870  }
871</script>
871
872
873  </body>
874</html>

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