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</div> 112 113 <div class="venue"> 114 115 <i>accepted</i> 116 Vietnam Journal of Mathematics 117 118 2025 119 </div> 120 121 122 123 124 125 126 127 128 129 <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> 134 135 136 137 138 139 140 141 142 143 144 145 </div> 146 147</div> 148 149<div class="publication" 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149 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2025-maxout","draft":false,"categories":[],"layout":"publication","year":2025,"title":"Maxout polytopes","status":"preprint","authors":["Andrei Balakin","Shelby Cox","Georg Loho","Bernd Sturmfels"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2509.21286","tags":["machine learning","polytopes","tropical geometry"],"highlight":true,"slug":"2025-maxout","ext":".html"}"> 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> 166 167 168 169 170 171 172 173 174 175 <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> 180 181 182 183 184 185 186 187 188 189 190 191 </div> 192 193</div> 194 195<h2 id="y2024" class="year">2024</h2> 196 197<div class="publication" data-pub="{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","excerpt":"","previous":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","excerpt":"","previous":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"
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 âfour clade rearrangementâ 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<meta property=\"og:description\" content=\"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 âfour clade rearrangementâ 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<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">
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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"id":"/publications/2023-regmixup","collection":"publications","next":{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","excerpt":"","previous":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"id":"/publications/2024-3sunlet","collection":"publications","next":{"path":"_publications/2024-bmtmMLD.html","relative_path":"_publications/2024-bmtmMLD.html","id":"/publications/2024-bmtmMLD","collection":"publications","url":"/publications/2024-bmtmMLD","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance","status":"accepted","authors":["Jane Coons","Shelby Cox","Aida Maraj","Ikenna Nometa"],"type":["research"],"venue":"
197Journal of Symbolic Computation, MEGA 2024 Special Edition","venue_tags":"JSC","pdf":"https://arxiv.org/abs/2402.10322","tags":["phylogenetics","ML degree"],"highlight":true,"slug":"2024-bmtmMLD","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Group-Based Phylogenetic Models on 3-Sunlet Networks | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"Group-Based Phylogenetic Models on 3-Sunlet Networks\",\"url\":\"https://www.shelbycox.github.io/publications/2024-3sunlet\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2024-3sunlet.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
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242 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2024-bmtmMLD","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance","status":"accepted","authors":["Jane Coons","Shelby Cox","Aida Maraj","Ikenna Nometa"],"type":["research"],"venue":"Journal of Symbolic Computation, MEGA 2024 Special Edition","venue_tags":"JSC","pdf":"https://arxiv.org/abs/2402.10322","tags":["phylogenetics","ML degree"],"highlight":true,"slug":"2024-bmtmMLD","ext":".html"}"> 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> 259 260 261 262 263 264 265 266 267 268 <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> 273 274 275 276 277 278 279 280 281 282 283 284 </div> 285 286</div> 287 288<div class="publication" data-pub="{"path":"_publications/2024-homaloidal.html","relative_path":"_publications/2024-homaloidal.html","excerpt":"","previous":{"path":"_publications/2024-bmtmMLD.html","relative_path":"_publications/2024-bmtmMLD.html","excerpt":"","previous":{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","excerpt":"","previous":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"id":"/publications/2024-3sunlet","collection":"publications","next":{"path":"_publications/2024-bmtmMLD.html","relative_path":"_publications/2024-bmtmMLD.html","id":"/publications/2024-bmtmMLD","collection":"publications","url":"/publications/2024-bmtmMLD","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance","status":"accepted","authors":["Jane Coons","Shelby Cox","Aida Maraj","Ikenna Nometa"],"type":["research"],"venue":"
288Journal of Symbolic Computation, MEGA 2024 Special Edition","venue_tags":"JSC","pdf":"https://arxiv.org/abs/2402.10322","tags":["phylogenetics","ML degree"],"highlight":true,"slug":"2024-bmtmMLD","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Group-Based Phylogenetic Models on 3-Sunlet Networks | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"Group-Based Phylogenetic Models on 3-Sunlet Networks\",\"url\":\"https://www.shelbycox.github.io/publications/2024-3sunlet\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2024-3sunlet.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
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<div class="venue"> 298 299 300 Algebraic Statistics 301 302 2024 303 </div> 304 305 306 307 308 309 310 311 312 313 <div class="extra-links"> 314 315 <a href="https://msp.org/astat/2024/15-2/p02.xhtml"> 316 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 317 </a> 318 319 320 321 322 323 324 325 326 327 328 329 </div> 330 331</div> 332 333<div class="publication" 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333 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2025-fwsafetyradius","draft":false,"categories":[],"layout":"publication","year":2025,"title":"Tropical Fermat-Weber Points over Bergman Fans","status":"accepted","authors":["Shelby Cox","John Sabol","Roan Talbut","Ruriko Yoshida"],"type":["research"],"venue":"Vietnam Journal of Mathematics","venue_tags":null,"pdf":"https://arxiv.org/abs/2505.09584","tags":["phylogenetics","tropical geometry"],"highlight":true,"slug":"2025-fwsafetyradius","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tropicalizing binary geometries | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tropicalizing binary geometries\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2024-tropbg\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2024-tropbg\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tropicalizing binary geometries\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"Tropicalizing binary geometries\",\"url\":\"https://www.shelbycox.github.io/publications/2024-tropbg\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2024-tropbg.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
333 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2024-tropbg","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Tropicalizing binary geometries","status":"published","authors":["Shelby Cox","Igor Makhlin"],"type":["research"],"venue":"Le Matematiche","venue_tags":null,"pdf":"https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/2744","tags":["physics","tropical geometry"],"highlight":true,"slug":"2024-tropbg","ext":".html"}"> 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 338 339 </div> 340 341 <div class="venue"> 342 343 <i>published</i> 344 Le Matematiche 345 346 2024 347 </div> 348 349 350 351 352 353 354 355 356 357 <div class="extra-links"> 358 359 <a href="https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/2744"> 360 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 361 </a> 362 363 364 365 366 367 368 369 370 371 372 373 </div> 374 375</div> 376 377<h2 id="y2023" class="year">2023</h2> 378 379<div class="publication" data-pub="{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","excerpt":"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.","previous":null,"id":"/publications/2019-eulerchar","collection":"publications","next":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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.\",\"headline\":\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\",\"url\":\"https://www.shelbycox.github.io/publications/2019-eulerchar\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023+-weightedfw.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
379 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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.\" />\n<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a 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379 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","excerpt":"","previous":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"id":"/publications/2023-regmixup","collection":"publications","next":{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","id":"/publications/2024-3sunlet","collection":"publications","url":"/publications/2024-3sunlet","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Group-Based Phylogenetic Models on 3-Sunlet Networks","status":"Open access","authors":["Shelby Cox","Elizabeth Gross","Samuel Martin"],"type":["research"],"venue":"Bulletin of Mathematical Biology","venue_tags":"BMB","pdf":"https://rdcu.be/eA9b3","tags":["tropical geometry","phylogenetics"],"highlight":true,"slug":"2024-3sunlet","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta 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379 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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 âfour clade rearrangementâ 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<meta property=\"og:description\" content=\"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 âfour clade rearrangementâ 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<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"
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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
379 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023+-weightedfw.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
379 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"}"> 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 384 385 </div> 386 387 <div class="venue"> 388 389 <i>preprint</i> 390 arXiv 391 392 2023 393 </div> 394 395 396 397 398 399 400 401 402 403 <div class="extra-links"> 404 405 <a href="https://arxiv.org/abs/2310.07732.pdf"> 406 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 407 </a> 408 409 410 411 412 413 414 415 416 417 418 419 </div> 420 421</div> 422 423<div class="publication" data-pub="{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","id":"/publications/2019-eulerchar","collection":"publications","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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.\" />\n<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"
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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"
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 âfour clade rearrangementâ 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<meta property=\"og:description\" content=\"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 âfour clade rearrangementâ 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<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">
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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023+-weightedfw.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
423 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","excerpt":"","previous":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"
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 âfour clade rearrangementâ 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<meta property=\"og:description\" content=\"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 âfour clade rearrangementâ 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<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">
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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"id":"/publications/2023-regmixup","collection":"publications","next":{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","excerpt":"","previous":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"id":"/publications/2024-3sunlet","collection":"publications","next":{"path":"_publications/2024-bmtmMLD.html","relative_path":"_publications/2024-bmtmMLD.html","id":"/publications/2024-bmtmMLD","collection":"publications","url":"/publications/2024-bmtmMLD","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance","status":"accepted","authors":["Jane Coons","Shelby Cox","Aida Maraj","Ikenna Nometa"],"type":["research"],"venue":"
423Journal of Symbolic Computation, MEGA 2024 Special Edition","venue_tags":"JSC","pdf":"https://arxiv.org/abs/2402.10322","tags":["phylogenetics","ML degree"],"highlight":true,"slug":"2024-bmtmMLD","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Group-Based Phylogenetic Models on 3-Sunlet Networks | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2024-3sunlet\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Group-Based Phylogenetic Models on 3-Sunlet Networks\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"Group-Based Phylogenetic Models on 3-Sunlet Networks\",\"url\":\"https://www.shelbycox.github.io/publications/2024-3sunlet\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2024-3sunlet.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
423 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2024-3sunlet","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Group-Based Phylogenetic Models on 3-Sunlet Networks","status":"Open access","authors":["Shelby Cox","Elizabeth Gross","Samuel Martin"],"type":["research"],"venue":"Bulletin of Mathematical Biology","venue_tags":"BMB","pdf":"https://rdcu.be/eA9b3","tags":["tropical geometry","phylogenetics"],"highlight":true,"slug":"2024-3sunlet","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\" />\n<meta name=\"author\" content=\"Dylan Cordaro\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-regmixup\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-regmixup\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Dylan Cordaro\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\",\"url\":\"https://www.shelbycox.github.io/publications/2023-regmixup\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-regmixup.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
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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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
423 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"}"> 424 <h3 id="/publications/2023-classtop">Classifying Tree Topology Changes along Tropical Line Segments</h3> 425 <div class="authors"> 426 <a href="shelbycox.github.io"><strong>Shelby Cox</strong></a> 427 428 </div> 429 430 <div class="venue"> 431 432 433 Algebraic Statistics 434 435 2023 436 </div> 437 438 439 440 441 442 443 444 445 446 <div class="extra-links"> 447 448 <a href="https://arxiv.org/pdf/2302.03611.pdf"> 449 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 450 </a> 451 452 453 454 455 456 457 458 459 460 461 462 </div> 463 464</div> 465 466<div class="publication" data-pub="{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","excerpt":"","previous":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta 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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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
466 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"id":"/publications/2023-regmixup","collection":"publications","next":{"path":"_publications/2024-3sunlet.html","relative_path":"_publications/2024-3sunlet.html","excerpt":"","previous":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","excerpt":"","previous":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical 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466 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2024-3sunlet","draft":false,"categories":[],"layout":"publication","year":2024,"title":"Group-Based Phylogenetic Models on 3-Sunlet Networks","status":"Open access","authors":["Shelby Cox","Elizabeth Gross","Samuel Martin"],"type":["research"],"venue":"Bulletin of Mathematical Biology","venue_tags":"BMB","pdf":"https://rdcu.be/eA9b3","tags":["tropical geometry","phylogenetics"],"highlight":true,"slug":"2024-3sunlet","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\" />\n<meta name=\"author\" content=\"Dylan Cordaro\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-regmixup\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-regmixup\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Dylan Cordaro\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification\",\"url\":\"https://www.shelbycox.github.io/publications/2023-regmixup\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-regmixup.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
466 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"}"> 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> 499 500 501 502 503 504 505 506 507 508 509 510 </div> 511 512</div> 513 514<h2 id="y2022" class="year">2022</h2> 515 516<div class="publication" data-pub="{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","excerpt":"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.","previous":null,"id":"/publications/2019-eulerchar","collection":"publications","next":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","id":"/publications/2019-eulerchar","collection":"publications","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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.\",\"headline\":\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\",\"url\":\"https://www.shelbycox.github.io/publications/2019-eulerchar\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","id":"/publications/2019-eulerchar","collection":"publications","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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.\" />\n<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","excerpt":"
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 âfour clade rearrangementâ 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","previous":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","id":"/publications/2023+-weightedfw","collection":"publications","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"id":"/publications/2023-classtop","collection":"publications","next":{"path":"_publications/2023-regmixup.html","relative_path":"_publications/2023-regmixup.html","id":"/publications/2023-regmixup","collection":"publications","url":"/publications/2023-regmixup","draft":false,"categories":[],"layout":"publication","year":2023,"title":"[¬Re] G-Mixup: Graph Data Augmentation for Graph Classification","authors":["Dylan Cordaro","Shelby Cox","Yiman Ren","Teresa Yu"],"type":"reproducibility","venue":"2022 Reproducibility Challenge","pdf":"https://github.com/ReScience/MLRC/blob/main/2022/34/article.pdf","venue_url":"https://paperswithcode.com/rc2022","tags":["machine learning","reproducibility"],"slug":"2023-regmixup","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Classifying Tree Topology Changes along Tropical Line Segments | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"
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 âfour clade rearrangementâ 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<meta property=\"og:description\" content=\"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 âfour clade rearrangementâ 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<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023-classtop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Classifying Tree Topology Changes along Tropical Line Segments\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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 âfour clade rearrangementâ 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.\",\"headline\":\"Classifying Tree Topology Changes along Tropical Line Segments\",\"url\":\"https://www.shelbycox.github.io/publications/2023-classtop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">
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 âfour clade rearrangementâ 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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023-classtop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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 âfour clade rearrangementâ 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","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023+-weightedfw.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
516 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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.\" />\n<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"}"> 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> 521 522 </div> 523 524 <div class="venue"> 525 526 527 Vietnam Journal of Mathematics 528 529 2022 530 </div> 531 532 533 534 535 536 537 538 539 540 <div class="extra-links"> 541 542 <a href="https://doi.org/10.1007/s10013-021-00526-3"> 543 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 544 </a> 545 546 547 548 549 550 551 552 553 <a href="https://www.doi2bib.org/bib/10.1007/s10013-021-00526-3"> 554 <i class="fas fa-book" aria-hidden="true"></i> Bibtex 555 </a> 556 557 558 559 560 </div> 561 562</div> 563 564<h2 id="y2019" class="year">2019</h2> 565 566<div class="publication" data-pub="{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","excerpt":"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.","previous":null,"id":"/publications/2019-eulerchar","collection":"publications","next":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","excerpt":"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.","previous":{"path":"_publications/2019-eulerchar.html","relative_path":"_publications/2019-eulerchar.html","excerpt":"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.","previous":null,"id":"/publications/2019-eulerchar","collection":"publications","next":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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.\",\"headline\":\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\",\"url\":\"https://www.shelbycox.github.io/publications/2019-eulerchar\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]},"id":"/publications/2022-treetop","collection":"publications","next":{"path":"_publications/2023+-weightedfw.html","relative_path":"_publications/2023+-weightedfw.html","excerpt":"","previous":{"path":"_publications/2022-treetop.html","relative_path":"_publications/2022-treetop.html","id":"/publications/2022-treetop","collection":"publications","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"id":"/publications/2023+-weightedfw","collection":"publications","next":{"path":"_publications/2023-classtop.html","relative_path":"_publications/2023-classtop.html","id":"/publications/2023-classtop","collection":"publications","url":"/publications/2023-classtop","draft":false,"categories":[],"layout":"publication","year":2023,"title":"Classifying Tree Topology Changes along Tropical Line Segments","authors":["Shelby Cox"],"type":["research"],"venue":"Algebraic Statistics","venue_tags":"Alg Stat","pdf":"https://arxiv.org/pdf/2302.03611.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023-classtop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>The tropical polytope is the set of all weighted tropical Fermat-Weber points | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<meta property=\"og:description\" content=\"I am a graduate student at the University of Michigan working with David Speyer.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2023+-weightedfw\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"I am a graduate student at the University of Michigan working with David Speyer.\",\"headline\":\"The tropical polytope is the set of all weighted tropical Fermat-Weber points\",\"url\":\"https://www.shelbycox.github.io/publications/2023+-weightedfw\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\n \n</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2023+-weightedfw.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length;
566 i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"","url":"/publications/2023+-weightedfw","draft":false,"categories":[],"layout":"publication","year":2023,"title":"The tropical polytope is the set of all weighted tropical Fermat-Weber points","status":"preprint","authors":["Shelby Cox","Mark Curiel"],"type":["research"],"venue":"arXiv","venue_tags":"arXiv","pdf":"https://arxiv.org/abs/2310.07732.pdf","tags":["tropical geometry","phylogenetics"],"highlight":false,"slug":"2023+-weightedfw","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Tree Topologies along a Tropical Line Segment | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<meta name=\"author\" content=\"Ruriko Yoshida\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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.\" />\n<meta property=\"og:description\" content=\"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
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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2022-treetop\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Tree Topologies along a Tropical Line Segment\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Ruriko Yoshida\"},\"description\":\"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.\",\"headline\":\"Tree Topologies along a Tropical Line Segment\",\"url\":\"https://www.shelbycox.github.io/publications/2022-treetop\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2022-treetop.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2022-treetop","draft":false,"categories":[],"layout":"publication","year":2022,"title":"Tree Topologies along a Tropical Line Segment","authors":["Ruriko Yoshida","Shelby Cox"],"doi":"10.1007/s10013-021-00526-3","type":["research"],"venue":"Vietnam Journal of Mathematics","pdf":"https://doi.org/10.1007/s10013-021-00526-3","tags":["phylogenetics","tropical geometry"],"slug":"2022-treetop","ext":".html"},"output":"<!DOCTYPE html>\n<html lang=\"en-US\">\n <head>\n <meta charset=\"utf-8\">\n <meta http-equiv=\"x-ua-compatible\" content=\"ie=edge\">\n <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\">\n <meta name=\"application-name\" content=\"Shelby Cox\">\n <meta name=\"theme-color\" content=\"#b00\">\n <link rel=\"shortcut icon\" href=\"/favicon.ico\"/>\n <link rel=\"icon\" type=\"image/png\" href=\"/favicon.png\" sizes=\"250x250\" />\n\n <!-- Begin Jekyll SEO tag v2.8.0 -->\n<title>Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams | Shelby Cox</title>\n<meta name=\"generator\" content=\"Jekyll v3.10.0\" />\n<meta property=\"og:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<meta name=\"author\" content=\"Shelby Cox\" />\n<meta property=\"og:locale\" content=\"en_US\" />\n<meta name=\"description\" content=\"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<meta property=\"og:description\" content=\"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.\" />\n<link rel=\"canonical\" href=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:url\" content=\"https://www.shelbycox.github.io/publications/2019-eulerchar\" />\n<meta property=\"og:site_name\" content=\"Shelby Cox\" />\n<meta property=\"og:type\" content=\"website\" />\n<meta name=\"twitter:card\" content=\"summary\" />\n<meta property=\"twitter:title\" content=\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\" />\n<script type=\"application/ld+json\">\n{\"@context\":\"https://schema.org\",\"@type\":\"WebPage\",\"author\":{\"@type\":\"Person\",\"name\":\"Shelby Cox\"},\"description\":\"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.\",\"headline\":\"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams\",\"url\":\"https://www.shelbycox.github.io/publications/2019-eulerchar\"}</script>\n<!-- End Jekyll SEO tag -->\n\n\n <link rel=\"alternate\" type=\"application/rss+xml\" title=\"Shelby Cox\" href=\"https://www.shelbycox.github.io/feed.xml\">\n\n <link href=\"https://use.fontawesome.com/releases/v6.2.1/css/all.css\" rel=\"stylesheet\">\n <link href=\"/styles.css\" rel=\"stylesheet\">\n\n \n</head>\n\n <body>\n \n<header class=\"page-header\">\n <nav class=\"container\">\n <a class=\"site-title\" href=\"/\">Shelby Cox</a>\n\n <a href=\"/projects/\" >Projects</a>\n <a href=\"/publications/\" aria-current=\"page\">Papers</a>\n <!-- <a href=\"/talks/\" >Talks</a> -->\n <!-- <a href=\"/blog/\" >Blog</a> -->\n <a href=\"/assets/CV_24112025.pdf\" >CV</a>\n\n <span class=\"external\">\n <a href=\"https://github.com/shelbycox\"><i class=\"fab fa-github\" aria-hidden=\"true\"></i> GitHub</a>\n </span>\n </nav>\n</header>\n\n\n <div class=\"page-content\">\n <section class=\"container \">\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</section>\n\n </div>\n\n <footer>\n <div class=\"container\">\n <div class=\"footer-col\">\n Hello from Ann Arbor.<br/>\n <abbr title=\"Last build on 2026-01-07\">January 2026</abbr>\n </div>\n <div class=\"footer-col\">\n <a href=\"https://github.com/shelbycox/shelbycox.github.io/edit/master/_publications/2019-eulerchar.html\">Edit this page</a>\n </div>\n </div>\n</footer>\n\n <script>\n function trim(str) {\n return str.replace(/^\\s+|\\s+$/g, '');\n }\n var headers = document.querySelectorAll(\"h2, h3, h4, h5, h6\");\n for (var i=0; i<headers.length; i++) {\n var h = headers[i];\n var name = h.getAttribute(\"id\");\n var title = h.innerHTML;\n h.innerHTML = '<a href=\"#' + name + '\" class=\"anchor\"><i class=\"fas fa-hashtag\"></i></a>' + trim(title);\n }\n</script>\n\n </body>\n</html>\n","content":"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.","url":"/publications/2019-eulerchar","draft":false,"categories":[],"layout":"publication","year":2019,"title":"Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams","authors":["Shelby Cox","Amal Mattoo"],"type":["research"],"venue":"Minnesota Journal of Undergraduate Mathematics","pdf":"https://pubs.lib.umn.edu/index.php/mjum/article/view/4137","slug":"2019-eulerchar","ext":".html","tags":[]}"> 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 571 572 </div> 573 574 <div class="venue"> 575 576 577 Minnesota Journal of Undergraduate Mathematics 578 579 2019 580 </div> 581 582 583 584 585 586 587 588 589 590 <div class="extra-links"> 591 592 <a href="https://pubs.lib.umn.edu/index.php/mjum/article/view/4137"> 593 <i class="far fa-file-pdf" aria-hidden="true"></i> PDF 594 </a> 595 596 597 598 599 600 601 602 603 604 605 606 </div> 607 608</div> 609 610<!--
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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>
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Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.