Commutative and Homological Algebra Market Presentations

Matthew Weaver is a postdoc at the University of Notre Dame, working with Claudia Polini. Before that, he was a graduate student at Purdue University, advised by Bernd Ulrich. Matthew's research interests lie at the intersection of commutative algebra and algebraic geometry. More specifically, he is interested in the study of blowing-up rings, namely Rees algebras of ideals and modules, and the implications these rings have within the study of syzygies, resolutions of singularities, and regularities of powers of ideals. Moreover, he is also interested in the applications of these rings to implicitization problems arising within the applied sciences, geometric modeling and computer-aided design in particular. As an educator, Matthew has taught numerous courses as a graduate student and as a postdoc. He hopes to continue to teach a wide range of classes and provide a welcoming learning environment for students.

Adam LaClair is a Ph.D. candidate at Purdue University advised by Professor Uli Walther. Adam's research interests lie in commutative algebra, combinatorics, and interactions between the two fields. Adam has obtained results on binomial edge ideals, Castelnuovo—Mumford regularity, singularity theory, and polytope theory. Adam has taught a variety of courses at the undergraduate level and was a teaching assistant for the MSRI/SLMath CMND 2023 Summer School.

Emanuela Marangone is a fifth-year Ph.D. student at the University of Notre Dame working under the supervision of Professor Juan Migliore. Her research interests lie in Commutative Algebra and Algebraic Geometry. In particular, she focuses on topics related to the Lefschetz Properties and the non-Lefschetz Locus. Her thesis concerns the non-Lefschetz locus for first cohomology modules of rank 2 vector bundles over P^2, extending the case of complete intersections of height 3. She also investigates analogous questions for higher-degree forms, studying the non-Lefschetz locus of conics. During her Ph.D. at the University of Notre Dame, Emanuela taught several undergraduate classes, both online and in person. She has also taught Linear Algebra to incarcerated college students as part of the Moreau College Initiative.

Alexandra (Sasha) Pevzner is a 5th year PhD student at the University of Minnesota, advised by Professor Victor Reiner. Her research interests lie at the intersection of commutative algebra and representation theory; in particular she enjoys using syzygies to understand modules which arise from group actions. Her thesis work studies a stability pattern in a certain family of ideals in invariant theory, whose homological data depends on the relationship between characteristic and the number of variables.

Monalisa Dutta is a Ph.D. candidate at the University of Kansas. Her advisor is Hailong Dao. Monalisa's research is in commutative algebra and its interactions with a variety of neighboring fields such as category theory, representation theory of algebras, and algebraic geometry. She has worked on exact categories and subfunctors of Ext, Ulrich modules, trace ideals, and module closure operations. In her capacity as an educator, Monalisa has been actively involved in teaching numerous courses, including Calculus, Precalculus, and College Algebra. Her experience extends to both being a teaching assistant and independently building a Calculus course. Her focus lies in promoting collaborative group work and establishing an interactive and inclusive learning atmosphere within the classroom.

Nikola Kuzmanovski is a mathematics PhD student at the University of Nebraska-Lincoln. His advisors are Jamie Radcliffe and Alexandra Seceleanu. Nikola's research is in commutative algebra, extremal combinatorics, and the connections between them. The focus is on Macaulay-Lex theorems, Hilbert functions, lex ideals, Macaulay posets, discrete isoperimetric inequalities, and intersection results like the Erdős-Ko-Rado theorem. Nikola had taught a variety of classes as an instructor of record and mentored undergraduate and graduate students.

Karthik Ganapathy is a graduate student at the University of Michigan, working with Andrew Snowden. He is interested in all things algebraic, and his focus has been on uncovering Noetherian phenomena in infinite variable polynomial rings using representation theory. His dissertation represents the first step in a program to extend Sam--Snowden's work on "GL-algebras" to positive characteristic. Karthik has been an Instructor of Record at Michigan for multiple sections of Calculus I and Precalculus and has experience teaching advanced undergraduate classes from his time in Chennai.

Trevor Arrigoni is a Ph.D. candidate at the University of Kansas, working with Daniel J. Hernández. Trevor's research involves measuring the behavior of singularities using Frobenius. In particular, he is interested in using ideas from integer programming, convex geometry, and number theory to develop explicit algorithms to compute the F-pure Threshold and Higher F-Thresholds of hypersurfaces. He aims to use his research to better understand the properties of these invariants. In the classroom, Trevor has taken on various roles, ranging from a teaching assistant to even designing his own calculus courses. As an educator, he believes that investing time in the small things goes a long way. In particular, he believes that building trust and rapport with his students is a powerful, yet simple, pedagogical tool. Because of this, Trevor is committed to creating a safe, welcoming environment which contributes to an enriched and fulfilling educational experience.

Sebastian Calvo is a \(5^{\textrm{th}}\) year PhD candidate at Pennsylvania State University. He studies algebraic geometry under the guidance of Jack Huizenga and John Lesieutre. His research area of interest involves studying configurations of hypersurfaces in the projective plane or projective 3-space. In particular, when a configuration exhibits a symmetry group, one can attempt to extract a lot of interesting properties of the arrangement. He has utilized the symmetry of a configuration's group to calculate its Waldschmidt constant, a numerical value that describes the speciality of the configuration and the calculation of which is an active area of research. Coming from a small liberal arts college, Sebastian is fiercely passionate about teaching and equity. He harvests a comfortable learning environment where students see learning as adventurous, asking questions as a learning tool and adapting when faced with difficulty. He has taken several mathematics education courses and he is a member of the mathematics teaching seminar, which discusses pedagogy and practical teaching strategies.

Cheng Meng is a PhD candidate at Purdue University. His advisors are Giulio Caviglia and Linquan Ma. Cheng's research interests are in commutative algebra and its interactions with algebraic geometry, homological algebra and combinatorics. He works on problems regarding the graded-irreducibility of graded modules, generic initial ideals, Boij-Soderberg theory for local cohomology tables, Lech's conjecture on multiplicities of flat local extension of Noetherian local rings, and singularities in prime characteristic. He has proved a new case of Lech's conjecture, and has found a way to decompose local cohomology tables of almost Cohen-Macaulay modules. As an instructor, he has taught an undergraduate class on calculus and was one of the official tutors in the ICTP summer school Graduate Course on Tight Closure of Ideals and its Applications.

Trung Chau is a Ph.D. candidate at the University of Utah. His advisors are Anurag K. Singh and Srikanth B. Iyengar. Trung works in a few different areas in Commutative Algebra. Recently, he has been focusing on determining the \(F\)-singularities of various naturally defined algebraic sets, and algorithmically constructing new free resolutions for monomial ideals over a polynomial ring using discrete Morse theory. Trung has taught a variety of undergraduate mathematics courses, at many different levels, both online and in-person. In the classroom, Trung emphasizes group work and active learning with students.

Le Tran is a Ph.D. candidate at New Mexico State University. His advisor is Professor Dr. Louiza Fouli. Le is interested in Commutative Algebra and its interactions with Combinatorics. His most recent result computes the depth of unicyclic graphs by using initially regular sequences on cycles. His favorite quote on teaching is "Education is not learning of facts but the training of the minds to think" by Albert Einstein, which inspires his goals in teaching and studying Mathematics. In his teaching, he emphasizes the importance of perseverance, as he firmly believes that studying Mathematics involves making mistakes and learning from them.

Sarah Poiani is a mathematics Ph.D. candidate at the University of New Mexico. Her advisor is Janet Vassilev and she completed her master's degree at UNM. Sarah's research is in Commutative Algebra. Specifically, she focuses mainly on pair operations, a generalization of closure operations: the duality between closure and interior operations and between cores and hulls, the interactions between various properties of general pair operations, and constructing new pair operations through ring extensions and collections. Sarah has taught several undergraduate mathematics courses, both online and in-person, and works to encourage a growth mindset in her students.