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56            Elo&iacute;sa Grifo | Research
57        </title>
58        <LINK REV="made" HREF="mailto:grifo&#x40;unl.edu">
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62<div class="main">
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64 <H1><center>
65            Research</center>
66        </H1>
67
68<center>
69<a href="https://arxiv.org/search/?query=Elo%C3%ADsa+Grifo&searchtype=author&abstracts=hide&order=-announced_date_first&size=50">
70  <img src="arxivicon.png" alt="arxiv" style="width:30px;height:30px;">
71</a>
72&nbsp;
73<a href="https://scholar.google.com/citations?user=RCz8ojQAAAAJ&hl=en&oi=ao">
74  <img src="scholaricon.png" alt="Google scholar" style="width:30px;height:30px;">
75</a>
76&nbsp;
77<a href="https://www.researchgate.net/profile/Eloisa_Grifo">
78  <img src="rgateicon.png" alt="researchgate" style="width:30px;height:30px;">
79</a>
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81
82<br>
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84<div align="justify"> 
85My research interests lie in commutative algebra and homological algebra. I am interested in applying homological algebra, characteristic p techniques, and p-derivations to different problems in commutative algebra. I have worked on questions that relate to (infinite) free resolutions, DG algebras, thick subcategories of \(D^b(R)\), cohomological support varieties, local cohomology, and symbolic powers. Here is a <a href=https://www.cim.pt/magazines/bulletin/42/article/298/pdf>general math audience article</a> article I wrote about symbolic powers, and here are lecture notes I am writing on <a href=SymbolicPowers.pdf>symbolic powers</a> (reader beware: these are under construction!). I wrote <a href=https://doi.org/10.18130/V3707WN5T>my PhD thesis</a> under the supervision of Craig Huneke, on the containment problem for symbolic and ordinary powers of ideals over regular rings. 
86
87</div>
88
89
90<div class="page-content">
91      <div class="wrapper">
92        <h2>Papers</h2>
93
94
95<div id="navcontainer">
96<ol>
97The embedded deformation problem for monomial ideals
98(with <a href=https://profiles.imperial.ac.uk/b.briggs>Ben Briggs</a> and <a href=https://www.joshpollitz.com>Josh Pollitz</a>)<br>
99<i>submitted</i>
100<div class=flex-container>
101<a href="https://arxiv.org/abs/2506.10827"><div class=arxiv>arXiv</div></a>
102&nbsp;
103<details><summary><div class="textstyle">abstract</div></summary>
104This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring R. More precisely, we establish that an embedded deformation of R corresponds exactly to a degree two central element in the homotopy Lie algebra of R, as well as a free summand of the conormal module of R. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine n-space that are the cohomological support of rings defined by n monomial relations where n is five or less.
105</details>      
106</div>
107We completely answer an old question of Avramov for the case of rings defined by monomial ideals: that embedded deformations of the ring R correspond to radical elements of degree 2 in the homotopy Lie algebra of R. We also study the cohomological support varieties of rings defined by monomial ideals, and classify all such varieties for ideals defined by up to 5 monomials.
108</li>
109
110<br><br>
111
112
113
114
115
116<li value="22"> Differential and symbolic powers of ideals (with <a href=https://www.dima.unige.it/~destefani/>Alessandro De Stefani</a> and <a href="https://jack-jeffries.github.io">Jack Jeffries</a>)<br>
117<i>submitted</i>
118<div class=flex-container>
119<a href="https://arxiv.org/abs/2503.21754"><div class=arxiv>arXiv</div></a>
120&nbsp;
121<details><summary><div class="textstyle">abstract</div></summary>
122We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of &#8484;-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a p-derivation &delta; in terms of &#8484;-linear differential operators and of &delta;. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of &#8484;-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no p-derivations.
123</details>      
124</div>
125We describe the symbolic powers of a prime ideal in a polynomial ring over any field in terms of differential operators; this extends a well-known result by removing a separability assumption. We give an analogous result in mixed characteristic: we describe the symbolic powers of prime ideals in a polynomial ring over a unramified DVR using differential operators and p-derivations.
126</li>
127
128<br>
129
130
131
132<li value="21">Bounds on cohomological support varieties (with <a href=https://profiles.imperial.ac.uk/b.briggs>Ben Briggs</a> and <a href=https://www.joshpollitz.com>Josh Pollitz</a>)<br>
133<i>Trans. Amer. Math. Soc. Ser. B</i> 11 (2024), 703-726
134<div class=flex-container>
135<a href="https://arxiv.org/pdf/2210.15574.pdf"><div class=arxiv>arXiv</div></a>
136&nbsp;
137<a href=https://www.ams.org/journals/btran/2024-11-21/S2330-0000-2024-00182-6/><div class=journalversion>journal</div></a>
138&nbsp;
139<details><summary><div class="textstyle">abstract</div></summary>
140Over a local ring \(R\), the theory of cohomological support varieties attaches to any bounded complex M of finitely generated R-modules an algebraic variety \(V_R(M)\) that encodes homological properties of M. We give lower bounds for the dimension of  \(V_R(M)\) in terms of classical invariants of R. In particular, when R is Cohen-Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M has finite projective dimension, we also give an upper bound for \(\textrm{dim } V_R(M)\) in terms of the dimension of the radical of the homotopy Lie algebra of R. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of \(R\). Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.
141</details>      
142</div>
143We show that if R is not a ci, then not every projective subvariety is the cohomological support variety of a complex over M. We do this by establishing upper and lower bounds on the dimension of cohomological supports of complexes over R.
144</li>
145
146<br>
147
148
149<li value="20">A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic (with <a href=https://www.dima.unige.it/~destefani/>Alessandro De Stefani</a> and <a href="https://jack-jeffries.github.io">Jack Jeffries</a>)<br>
150<i>Mathematische Zeitschrift</i> 301 (2022), no. 4, 4141-4151.
151<div class=flex-container>
152<a href="https://arxiv.org/abs/2104.00612"><div class=arxiv>arXiv</div></a>
153&nbsp;
154<a href=https://rdcu.be/cOIDI><div class=journalversion>journal</div></a>
155&nbsp;
156<details><summary><div class="textstyle">abstract</div></summary>
157We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
158</details>      
159</div>
160Using p-derivations and differential operators, we prove an explicit uniform Chevalley Lemma for direct summands of polynomial rings in mixed characteristic.
161</li>
162
163<br>
164
165
166
167<li value="19">Constructing non-proxy small test modules for the complete intersection property (with <a href=https://profiles.imperial.ac.uk/b.briggs>Ben Briggs</a> and <a href=https://www.joshpollitz.com>Josh Pollitz</a>)
168<br>
169<i>Nagoya Mathematical Journal</i>, Volume 246, June 2022, pp. 412 - 429
170<div class="flex-container">
171<a href=https://arxiv.org/abs/2009.11800><div class=arxiv>arXiv</div></a>
172&nbsp;
173<a href=https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/constructing-nonproxy-small-test-modules-for-the-complete-intersection-property/E8CCDBC8E369F18949659FDD15EC3FFF><div class=journalversion>journal</div></a>
174&nbsp;
175<details><summary><div class="textstyle">abstract</div></summary>
176A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category D<sup>f</sup>(R), which contains only small objects if and only if R is regular.
177<br>  
178Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in D<sup>f</sup>(R) is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.
179</details>
180</div>
181Given a local ring that is not a complete intersection, we aim to construct finitely generated modules that are not proxy small, completing a characterization of CIs given by Pollitz. We succeed in this goal for a large class of rings that includes all equipresented rings.
182<br>
183</li>
184
185
186<br>
187
188
189<li value="18">Demailly's conjecture and the containment problem (with <a href=https://sites.google.com/view/sankhaneelbisui/home>Sankhaneel Bisui</a>, <a href=http://www.math.tulane.edu/~tai/>T&agrave;i Huy H&agrave;</a>, and <a href=https://sites.google.com/view/thainguyenmath/home>Th&aacute;i Th&agrave;nh Nguy&#7877;n</a>)
190<br>
191<i>Journal of Pure and Applied Algebra</i>, Volume 226, Issue 4, 2022.
192<div class="flex-container">
193<a href=https://arxiv.org/pdf/2009.05022.pdf><div class=arxiv>arXiv</div></a>
194&nbsp;
195<a href=https://www.sciencedirect.com/science/article/pii/S0022404921002048?via%3Dihub><div class=journalversion>journal</div></a>
196&nbsp;
197<details><summary><div class="textstyle">abstract</div></summary>
198We investigate Demailly's Conjecture for a general set of sufficiently many
199points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing
200a lower bound for the Waldschmidt constant of a set of points in projective
201spaces. We also study a containment between symbolic and ordinary powers
202conjectured by Harbourne and Huneke that in particular implies Demailly's
203bound, and prove that a general version of that containment holds for
204determinantal ideals and defining ideals of star configurations.
205</details> 
206</div> 
207We show each of Demailly's inequalities holds for sufficiently many general points in 
208&#x2119;<sup>n</sup>, and show that the Harbourne-Huneke containment that would imply Demailly's bound holds for star configurations and generic determinantal ideals.
209</li>
210
211<br>
212
213
214<li value="17">Expected resurgence of ideals defining Gorenstein rings (with <a href="http://www.math.virginia.edu/people/clh4xd">Craig Huneke</a> and <a href=https://math.virginia.edu/people/vm6y/>Vivek Mukundan</a>)
215<br>
216<i>Michigan Math Journal</i> 73 (2023), no. 4, 735-749.
217<div class="flex-container">
218<a href="http://arxiv.org/abs/2007.12051"><div class=arxiv>arXiv</div></a>
219&nbsp;
220<details><summary><div class="textstyle">abstract</div></summary>
221Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.
222</details>   
223</div>   
224We show that when R is a regular ring containing a field, and R/I is Gorenstein, then I has expected resurgence (and thus satisfies Stable Harbourne) as long as R has prime characteristic or the symbolic Rees algebra of I is noetherian.
225<br>
226</li>
227
228
229<br>
230
231
232
233<li value="16">Chudnovsky's Conjecture and the stable Harbourne&mdash;Huneke containment (with <a href=https://sites.google.com/view/sankhaneelbisui/home>Sankhaneel Bisui</a>, <a href=http://www.math.tulane.edu/~tai/>T&agrave;i Huy H&agrave;</a>, and <a href=https://sites.google.com/view/thainguyenmath/home>Th&aacute;i Th&agrave;nh Nguy&#7877;n</a>)
234<br>
235<i>Trans. Amer. Math. Soc. Ser. B</i> 9 (2022), 371-394 
236<div class=flex-container>
237<a href="http://arxiv.org/abs/2004.11213"><div class=arxiv>arXiv</div></a>
238&nbsp
239<a href=https://www.ams.org/journals/btran/2022-09-12/S2330-0000-2022-00103-5/S2330-0000-2022-00103-5.pdf><div class=journalversion>journal</div></a>
240&nbsp;
241<details><summary><div class="textstyle">abstract</div></summary>
242In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky's Conjecture and the stable version of the Harbourne&mdash;Huneke containment conjectures for a general set of sufficiently many points.
243</details>
244</div>     
245We show Chudnovsky's Conjecture for sufficiently large sets of general points, by giving sufficient conditions on the stable versions of the Harbourne&mdash;Huneke conjectures.
246</li>
247
248<br>
249
250<li value="15">Expected resurgences and symbolic powers of ideals (with <a href="http://www.math.virginia.edu/people/clh4xd">Craig Huneke</a> and <a href=https://math.virginia.edu/people/vm6y/>Vivek Mukundan</a>)
251<br>
252<i>Journal of the London Mathematical Society</i> (2) 102 (2020) 453-469.
253
254<div class="flex-container">
255<a href="https://arxiv.org/abs/1903.12122"><div class=arxiv>arXiv</div></a> &nbsp;
256<a href=https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12324><div class=journalversion>journal</div></a> &nbsp;
257<details><summary><div class="textstyle">abstract</div></summary>
258We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.
259</details> 
260
261</div>
262
263
264We find sufficient conditions for stable Harbourne, and show they apply to certain classes.
265<br>     
266</li>
267
268<br>
269
270<li value="14">A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
271<br>
272<i>J. Pure Appl. Algebra</i>, Volume 224, Issue 12, 2020.
273<div class="flex-container">
274<a href="https://arxiv.org/abs/1809.06955"><div class=arxiv>arXiv</div></a>
275&nbsp;
276<a href=https://www.sciencedirect.com/science/article/pii/S0022404920301341?via%3Dihub><div class=journalversion>journal</div></a>
277&nbsp;
278<details><summary><div class="textstyle">abstract</div></summary>
279The symbolic powers I<sup>(n)</sup> of a radical ideal I in a polynomial ring
280consist of the functions that vanish up to order n in the variety defined by
281I. These do not necessarily coincide with the ordinary algebraic powers
282I<sup>n</sup>, but it is natural to compare the two notions. The containment problem
283consists of determining the values of $n$ and $m$ for which I<sup>(n)</sup> &sube; I<sup>m</sup> holds. When I is an ideal of height 2 in a regular ring, I<sup>(3)</sup> &sube; I<sup>2</sup>  may fail, but we show that this containment does hold for the
284defining ideal of the space monomial curve (t<sup>a</sup>, t<sup>b</sup> t<sup>c</sup>). More generally,
285given a radical ideal I of big height h, while the containment
286I<sup>(hn-h+1)</sup> &sube; I<sup>n</sup> conjectured by Harbourne does not necessarily hold
287for all n, we give sufficient conditions to guarantee such containments for
288n sufficiently large.
289</details>  
290</div>  
291This is a two part paper: the first half introduces the stable Harbourne Conjecture and provides some evidence; the second half shows that the defining ideals I of space monomial curves (t<sup>a</sup>,t<sup>b</sup>,t<sup>c</sup>) satisfy I<sup>(3)</sup> &sube; I<sup>2</sup>.  
292</li>
293
294<br>
295
296<li value="13">Symbolic power containments in singular rings in positive characteristic (with <a href=https://www.math.purdue.edu/~ma326/>Linquan Ma</a> and <a href=https://www.math.utah.edu/~schwede/>Karl Schwede</a>)
297<br>
298<i>manuscripta mathematica</i> 170 (2023), no. 3-4, 471-496. 
299<div class=flex-container>
300<a href="https://arxiv.org/abs/1911.06307"><div class=arxiv>arXiv</div></a>
301&nbsp;
302<a href=https://link.springer.com/article/10.1007/s00229-021-01359-7><div class=journalversion>journal</div></a>
303&nbsp;
304<details><summary><div class="textstyle">abstract</div></summary>
305The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I<sup>(a)</sup> &sube; I<sup>b</sup>. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals &mdash; although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.
306</details> 
307</div>     
308We extend Fedder's Criterion to ideals of finite projective dimension over Gorenstein F-finite rings, and use it to extend my results with Huneke on symbolic powers. 
309</li>
310
311<br>
312
313<li value="12">A Zariski-Nagata Theorem for smooth &#x2124;-algebras (with <a href=https://www.dima.unige.it/~destefani/>Alessandro De Stefani</a> and <a href="https://jack-jeffries.github.io">Jack Jeffries</a>)<br>
314<i>J. Reine Angew. Math.</i> (Crelles Journal) 2020 (761), 123&mdash;140. 
315<div class=flex-container>
316<a href="https://arxiv.org/abs/1709.01049"><div class=arxiv>arXiv</div></a>
317&nbsp;
318<a href=https://www.degruyter.com/view/j/crll.ahead-of-print/crelle-2018-0012/crelle-2018-0012.xml?format=INT><div class=journalversion>journal</div></a>
319&nbsp;
320<details><summary><div class="textstyle">abstract</div></summary>In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to Commutative Algebra. 
321</details>      
322</div>
323We give a mixed characteristic version of the Zariski&mdash;Nagata Theorem, mixing both differential operators and p-derivations. 
324</li>
325
326<br>
327
328<li value="11">Symbolic powers of ideals defining F-pure and strongly F-regular rings (with <a href="http://www.math.virginia.edu/people/clh4xd">Craig Huneke</a>) <br>
329<I>Int. Math. Res. Not. (IMRN) </i> 2019, no. 10, 2999&mdash;3014.
330<div class=flex-container>
331<a href="https://arxiv.org/abs/1702.06876"><div class=arxiv>arXiv</div></a>
332&nbsp;
333<a href=https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnx213/4124876/Symbolic-Powers-of-Ideals-Defining-F-pure-and><div class=journalversion>journal</div></a>
334&nbsp;
335<details><summary><div class="textstyle">abstract</div></summary>
336Given a radical ideal I in a regular ring R, the containment problem of symbolic and ordinary powers of I consists of determining when the containment I<sup>(a)</sup> &sube; I<sup>b</sup> holds. By work of Ein-Lazersfeld-Smith and Hochster-Huneke, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when R/I is F-pure, and prove tighter containments in the case when R/I is strongly F-regular.</details>      
337</div>
338We show that over a regular ring R of characteristic p, if R/I is F-pure, then I satisfies Harbourne's Conjecture. We also show a stronger statement holds if R/I is strongly F-regular.
339</li>
340
341<br>
342
343<li value="10">Local cohomology and Lyubeznik numbers of F-pure rings (with <a href=https://www.dima.unige.it/~destefani/>Alessandro De Stefani</a> and <a href="http://www.cimat.mx/~luisnub/home.html">Luis N&uacute;&ntilde;ez-Betancourt</a>)
344<br>
345<i>J. Algebra</i> 571 (2021), 316-338.
346<br>
347<div class=flex-container>
348<a href="https://arxiv.org/abs/1710.00462"><div class=arxiv>arXiv</div></a> &nbsp;
349<a href=https://www.sciencedirect.com/science/article/pii/S0021869318304265><div class=journalversion>journal</div></a>
350&nbsp;
351<details><summary><div class="textstyle">abstract</div></summary>
352In this article, we study certain local cohomology modules over F-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the F-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally F-split projective variety. 
353</details>
354</div>
355We give a formula for the Lyubeznik numbers of a standard graded F-pure ring, and show that the associated primes of certain Ext-modules are all compatible ideals.
356</li>
357
358<br>
359
360<li value="9">Lower bounds on projective levels of complexes (with Hannah Altmann, <a href="https://www.math.ku.edu/~j402m915/">Jonathan Monta&ntilde;o</a>, <a href="https://www.math.ntnu.no/~williats/">William Sanders</a>, and <a href="http://www.math.unl.edu/~tvu5/">Thanh Vu</a>) <br>
361<i>Journal of Algebra</i>, 491C (2017), pp. 343-356. 
362<div class=flex-container>
363<a href="http://arxiv.org/abs/1512.08534"><div class=arxiv>arXiv</div></a> &nbsp;
364<a href="http://www.sciencedirect.com/science/article/pii/S0021869317304581"><div class=journalversion>journal</div></a>
365&nbsp;
366<details><summary><div class="textstyle">abstract</div></summary>
367For an associative ring R, the level of a perfect complex F is the smallest number of mapping cones needed to build F from add(R). We establish lower bounds of the level of F in terms of the vanishing of homology of F. We then use these bounds to derive an improved version of The New Intersection Theorem for level when R is a commutative local ring containing a field.
368</details> 
369</div>
370We give some lower bounds on the projective levels and derive a new version of The New Intersection Theorem for levels.     
371</li>
372
373<br>
374
375<li value="8">On the growth of deviations (with <a href="http://home.sandiego.edu/~aboocher/">
375Adam Boocher<a/>, <a href="http://www.dima.unige.it/~dali/">Alessio D'Al&igrave;</a>, <a href="https://www.math.ku.edu/~j402m915/">Jonathan Monta&ntilde;o</a>, and <a href="https://sites.google.com/site/alessiosammartano/">Alessio Sammartano</a>) 
376<br>
377<i>Proceedings of the American Mathematical Society</i> 144 (2016), pp. 5049-5060.
378<div class=flex-container>
379<a href="http://arxiv.org/abs/1504.01066"><div class=arxiv>arXiv</div></a>
380&nbsp;
381<a href="http://www.ams.org/journals/proc/0000-000-00/S0002-9939-2016-13132-8/S0002-9939-2016-13132-8.pdf"><div class=journalversion>journal</div></a>
382&nbsp;
383<details><summary><div class="textstyle">abstract</div></summary>
384The deviations of a graded algebra are a sequence of integers that determine the Poincar&eacute; series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.
385</details> 
386</div>
387We prove some results on deviations that resemble well-known facts about the Betti numbers; for example, that the deviations of algebras with a fixed Hilbert series are maximized by the Lex-segment ideal.  
388</li>
389
390<br>
391
392<li value="7">Edge ideals and DG algebra resolutions (with <a href="http://home.sandiego.edu/~aboocher/">Adam Boocher<a/>, <a href="http://www.dima.unige.it/~dali/">Alessio D'Al&igrave;</a>, <a href="https://www.math.ku.edu/~j402m915/">Jonathan Monta&ntilde;o</a>, and <a href="https://sites.google.com/site/alessiosammartano/">
392Alessio Sammartano</a>) <br>
393<i>Le Matematiche </i> 70 (2015), no.1, 215-237. 
394<div class=flex-container>
395<a href="http://arxiv.org/abs/1504.01450"><div class=arxiv>arXiv</div></a> 
396&nbsp;
397<a href="http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/1236"><div class=journalversion>journal</div></a>
398&nbsp;
399<details><summary><div class="textstyle">abstract</div></summary>
400Let R=S/I where S=k[T_1,&#8230;,T_n] and I is a homogeneous ideal in S. The acyclic closure R of k over R is a DG algebra resolution obtained by means of Tate's process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model S[X], a DG algebra resolution of R over S. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when I is the edge ideal of a path or a cycle. We determine the behavior of the deviations &epsilon;_i(R), which are the number of variables in R in homological degree i. We apply our results to the study of the k-algebra structure of the Koszul homology of R.
401</details> 
402</div>
403We study the deviations of certain edge ideals, and the algebra structure of their Koszul homology.     
404</li>
405
406
407<br>
408
409
410
411<h3>Surveys and papers that are mostly expository</h3>
412
413
414<li value="6">Symbolic powers
415<br>
416<div class="flex-container">
417<a href=https://www.cim.pt/magazines/bulletin/42/article/298/pdf><div class=journalversion>journal</div></a>
418&nbsp;
419<details><summary><div class="textstyle">abstract</div></summary>
420Symbolic powers arise naturally in commutative algebra from the theory of primary decomposition, but they also contain geometric information, thanks to a classical result of Zariski and Nagata. Computing primary decompositions is a difficult computational problem, and as a result, many natural questions about symbolic powers remain wide open. We will briefly introduce symbolic powers and describe some of the main open problems on the subject, and point to some recent research advances.
421</details>
422</div>
423This is a survey on symbolic powers aimed at a general math audience.
424<br>
425</li>
426
427<br>
428
429
430<li value="5">Lower bounds on Betti numbers (with <a href="http://home.sandiego.edu/~aboocher/">Adam Boocher</a>)
431<br>
432In: Peeva I. (eds) Commutative Algebra. Springer, Cham.
433<div class="flex-container">
434<a href=https://arxiv.org/abs/2108.05871><div class=arxiv>arXiv</div></a>
435&nbsp;
436<a href=https://link.springer.com/chapter/10.1007/978-3-030-89694-2_2><div class=journalversion>journal</div></a>
437<details><summary><div class="textstyle">abstract</div></summary>
438We survey recent results on bounds for betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and betti numbers, and discuss some of the reasons why one would study these.
439</details>
440</div>
441This is a survey on lower bounds for Betti numbers over polynomial rings.
442<br>
443</li>
444
445<br>
446
447
448<li value="4">Symbolic Rees Algebras (with <a href="http://www.math.unl.edu/~aseceleanu2/">Alexandra Seceleanu</a>)
449<br>
450In: Peeva I. (eds) Commutative Algebra. Springer, Cham.
451<div class="flex-container">
452<a href=https://arxiv.org/abs/2012.01617><div class=arxiv>arXiv</div></a>
453&nbsp;
454<a href=https://link.springer.com/chapter/10.1007/978-3-030-89694-2_11><div class=journalversion>journal</div></a>
455&nbsp;
456<details><summary><div class="textstyle">abstract</div></summary>
457We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.
458</details>
459</div>
460This is a survey on symbolic Rees algebras.
461<br>
462</li>
463
464<br>
465
466<li value="3"> Symbolic powers of ideals (with <a href="https://www.math.ku.edu/~hdao/">Hailong Dao</a>, <a href=https://www.dima.unige.it/~destefani/>Alessandro De Stefani</a>, <a href="http://www.math.virginia.edu/people/clh4xd">Craig Huneke</a> and <a href="http://www.cimat.mx/~luisnub/home.html">Luis N&uacute;&ntilde;ez-Betancourt</a>) <br>
467in <i>Singularities and foliations. Geometry, topology and applications</i>, pp. 387-432, Springer Proc. Math. Stat., 222, Springer, Cham, 2018.
468<div class=flex-container>
469<a href="https://arxiv.org/abs/1708.03010"><div class=arxiv>arXiv</div></a>
470&nbsp;
471<a href=https://link.springer.com/chapter/10.1007/978-3-319-73639-6_13><div class=journalversion>journal</div></a>
472&nbsp;
473<details><summary><div class="textstyle">abstract</div></summary>We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.
474</details>
475</div>
476The title says it all: this is a survey on symbolic powers of ideals.
477</li>
478
479
480<br>
481
482<h3>Papers on mathematical software</h3>
483
484
485<li value=2>Computations involving symbolic powers (with <a href=http://www.math.unl.edu/~bdrabkin2/>Ben Drabkin</a>, <a href="http://www.math.unl.edu/~aseceleanu2/">Alexandra Seceleanu</a>, and <a href="https://people.hamilton.edu/bstone/index.html">Branden Stone<a/>)
486<br>
487<i>Journal of Software for Algebra and Geometry</i> 9 (2019), 71&mdash;80.
488<div class=flex-container>
489<a href="https://arxiv.org/abs/1712.01440"><div class=arxiv>arXiv</div></a>
490&nbsp;
491<a href=https://msp.org/jsag/2019/9-1/jsag-v9-n1-p09-s.pdf><div class=journalversion>journal</div></a>
492&nbsp;
493<details><summary><div class="textstyle">abstract</div></summary>
494Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously difficult to compute, and there are seemingly simple questions related to symbolic powers that remain open even over polynomial rings. In this paper, we describe a Macaulay2 software package that allows for computations of symbolic powers of ideals and which can be used to study the equality and containment problems, among others.
495</details> 
496</div>
497This paper describes the Macaulay2 package SymbolicPowers.     
498</li>
499
500<br>
501
502<li value=1>The software package SpectralSequences (with <a href="http://home.sandiego.edu/~aboocher/">Adam Boocher<a/> and <a href="http://www.math.unb.ca/~ngrieve/">Nathan Grieve</a>)
503<br>
504<i>submitted</i> 
505<div class=flex-container>
506<a href="https://arxiv.org/abs/1610.05338"><div class=arxiv>arXiv</div></a>
507&nbsp;
508<details><summary><div class="textstyle">abstract</div></summary>We describe the computer algebra software package <i>SpectralSequences</i> for the computer algebra system <i>Macaulay2</i>. This package implements many data types, objects and algorithms which pertain to, among other things, filtered complexes, spectral sequence pages and maps therein. We illustrate some of the syntax and capabilities of SpectralSequences by way of several examples.
509</details> 
510</div>
511This paper describes the Macaulay2 package SpectralSequences.     
512</li>
513
514</ol>
515</div>
516
517<br><br><br>
518
519 <div class="page-content">
520      <div class="wrapper">
521        <h2 id="notes"><a href=talks.html>Talks and notes</a></h2>
522
523Here are my <a href=SymbolicPowers.pdf>notes on symbolic powers</a>, and other <a href=talks.html>notes, videos, and slides</a> from talks I have given.
524
525
526<div class="talks">
527<div class="talks1">
528<iframe src="https://player.vimeo.com/video/904293781?h=cc30db3f1b" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>
529
530<a href="https://vimeo.com/904293781">The Dimension of Cohomological Support Varieties</a>
531</div>
532
533<div class="talks2">
534<iframe width="600" height="410" src="https://www.youtube.com/embed/Be83xC1isy4" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
535<center>virtual talk about symbolic powers</center>
536</div></div>
537<br>
538
539<!--For shorter notes, see my <a href=http://ac2019.eventos.cimat.mx>CIMAT lectures</a>, which are available in:
540<br>
541<br>
542<a href="SymbolicPowersCIMAT.pdf">english version</a> 
543&nbsp;
544<a href="SymbolicPowersEspanol.pdf">versi&oacute;n en espa&ntilde;ol</a> 
545&nbsp;
546<a href="SymbolicPowersTuga.pdf">vers&atilde;o em portugu&ecirc;s</a>
547
548
549<br><br>
550
551<a href=http://www.math.utah.edu/awmchapter/conference/>BRIDGES</a> notes, aimed at advanced undergraduates:
552<br><br>
553
554<a href=bridges/bridges.pdf>Lecture notes</a>
555&nbsp; &nbsp;
556<a href=bridges/BridgesProblems.pdf>Problem list</a>
557&nbsp; &nbsp;
558<a href=bridges/bridges1.pdf>Day 1</a>
559&nbsp; &nbsp;
560<a href=bridges/bridges2.pdf>Day 2</a>
561&nbsp; &nbsp;
562<a href=bridges/bridges3.pdf>Day 3</a>
563--!>
564
565
566
567
568
569
570<div class="page-content">
571      <div class="wrapper">
572        <h2 id=software>Software</h2>
573
574I often use <a href=http://www.math.uiuc.edu/Macaulay2/>Macaulay2</a> in my research, and I have contributed to a few Macaulay2 packages, some of which you can find on <a href=https://github.com/eloisagrifo>my GitHub page</a>.
575
576<br><br>
577
578  <li>
578<a href="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.10/share/doc/Macaulay2/SpectralSequences/html/">SpectralSequences</a> <br>(developed with <a href="http://www.math.berkeley.edu/~daffyd">David Berlekamp</a>, <a href="http://home.sandiego.edu/~aboocher/">Adam Boocher<a/>, <a href="http://www.math.unb.ca/~ngrieve">Nathan Grieve</a>, <a href="http://www.mast.queensu.ca/~ggsmith">Gregory G. Smith</a>, and <a href="http://www.math.unl.edu/~tvu5/">Thanh Vu</a>) <br>
579This package comes with Macaulay2 version &#8805; 1.10.
580</li>
581<ol>
582    This is a <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2</a> package that provides tools for effective computation of the pages and differentials in spectral sequences obtained from many kinds of filtered chain complexes. <br><br>
583  </ol>
584</ul>
585
586
587  <li><a href="https://github.com/eloisagrifo/SymbolicPowers">SymbolicPowers</a> <br>(with contributions from <a href=http://www.math.unl.edu/~bdrabkin2/>Ben Drabkin</a>, <a href="http://www.math.unl.edu/~aseceleanu2/">Alexandra Seceleanu</a>, and <a href="https://people.hamilton.edu/bstone/index.html">Branden Stone<a/>)<br>
588<a href=https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.16/share/doc/Macaulay2/SymbolicPowers/html/>Version 2.0</a> comes with Macaulay2 version \(\geqslant\) 1.14.
589<ol>
590    This is a <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2</a> package that provides tools for computing symbolic powers.
591<br><br>
592  </ol>
593</li>
594
595
596<li><a href="https://github.com/eloisagrifo/levels">ThickSubcategories</a> <br>(with <a href=https://www.math.uni-bielefeld.de/~jletz/>Janina Letz</a> and <a href=https://www.math.utah.edu/~pollitz/>Josh Pollitz</a>)<br>
597Under construction; use with caution.
598<ol>
599    This is a <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2</a> package that provides tools for computations related to thick subcategories of D(R); for example, it computes (or gives bounds for) levels of modules and complexes, and produces modules that are non-proxy small over non-ci quotients of polynomial rings.
600<br><br>
601  </ol>
602</li>
603</ul>
604
605<br>
606<br>
607
608<center>
609<img src=NSFlogo.png height=100 hspace=25>
610<br>
611My research is supported by 
612<br>
613<br>
614NSF grant <a href=https://www.nsf.gov/awardsearch/showAward?AWD_ID=2140355&HistoricalAwards=false>DMS-2140355</a>
615<br>
616and
617<br>
618NSF CAREER grant <a href=https://www.nsf.gov/awardsearch/showAward?AWD_ID=2236983&HistoricalAwards=false>DMS-2236983</a>.
619</center>
620
621
622
623<br>
624<hr color=#808080>
625<p class="caption"><center>
626	email: grifo&#64;unl.edu &nbsp; | &nbsp; Office: 339 Avery Hall
627</center> </p>
628</div>
629
630</BODY>
631
632
633
634
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