Spring 2024
Time & Location: All talks are on Wednesday in Richardson Building - RB-117 (G), at 3:00 PM unless otherwise noted.
Organizer: Daniel Bernstein
Information on up coming events can be found at unofficial seminar website: Here
October 2
Title: Explicit Hypergeometric Modularity and Applications
Speaker: Brian Grove - LSU
Abstract: The existence of hypergeometric motives predicts that hypergeometric Galois representations are modular. More precisely, explicit identities between special values of hypergeometric character sums and coefficients of certain newforms on appropriate arithmetic progressions of primes are expected. I will discuss a general method to prove these hypergeometric modularity results in dimensions two and three. Then I will use this method to explore new connections between hypergeometric functions and modular forms in the complex and p-adic settings. This is joint work with Michael Allen, Ling Long, and Fang-Ting Tu.
October 9
Title: Reconstructing configurations and graphs from unlabeled distance measurements
Speaker: Shlomo Gortler - Harvard
Abstract: Place a configuration of n points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph
determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence). This question is motivated by a number of inverse problem applications. In this talk, I will talk about what is known about this question.
October 16
Title: Symbolic Powers of Matroids
Speaker: Vinh Nguyen - University of Arkansas
Abstract: In general, it is quite hard to explicitly describe the minimal generators of the symbolic powers of any class of ideals, even in the case of square-free monomial ideals. In recent work with Paolo Mantero, we provide a structure result on the minimal generators of symbolic powers of a class of square-free monomial ideals that come from matroids. Matroids are combinatorial structures which abstract the structure of linear independence of vectors. Their Stanley-Reisner ideals have nice properties. For instance, every symbolic power is Cohen-Macaulay. In fact, they are the only square-free monomial ideals for which every symbolic power is Cohen-Macaulay.
In this talk I will introduce symbolic powers, matroids and their related ideals, and discuss our structure result along with various applications. If time permits, I would also like to talk about the minimal resolution of the symbolic powers of matroids. It turns out that their Betti numbers are supported on their symbolic powers. In fact, this is yet another characterization of matroids; they are the only square-free monomial ideals where this is true.
October 23
Title: Newton non-degenerate ideals in regular local domains
Speaker: Vinh Pham - Tulane
Abstract: The concept of Newton non-degenerate (NND) ideals in rings of holomorphic germs was introduced by M. J. Saia in 1996 to understand geometric invariants of complex-valued functions with an isolated singularity. We extend this notion to regular local domains and investigate algebraic invariants and properties of graded families of NND ideals in terms of associated convex bodies. This is joint work with Tai Huy Ha and Thai Thanh Nguyen.
October 30
Title: Asymptotic regularity and depth of invariant chains of edge ideals
Speaker: Hop D. Nguyen - Institute of Mathematics, Vietnam Academy of Science and Technology
Abstract: We consider the asymptotic behavior of chains of monomial ideals that are stable under the action of the monoid Inc of increasing functions N → N. It is conjectured that for such chains, the regularity and projective dimension are eventually linear functions. We confirm the conjecture and provide complete description of the regularity and projective dimension (equivalently, the depth) in the case of chains of edge ideals. Remarkably, if the ideals in the chain are non-zero, then the regularity function is eventually constant with only two possible limiting values, and the same thing happens for the depth. Our results and their proofs also reveal many interesting combinatorial and topological properties of Inc-invariant chains of graphs and their independence complexes. Joint work with Tran Quang Hoa, Do Trong Hoang, Dinh Van Le, and Thái Thành Nguyễn.
Location: Richardson Building - RB-117 (G)
November 6
Title: Tropicalization in Combinatorics
Speaker: Greg Blekherman - Georgia Tech
Abstract: I will survey some recent applications of tropicalization in combinatorics. Tropicalization captures possible orders of growth of counted quantities (such as number of certain subgraphs of a graph, or the number matroids of certain rank). This provides a coarse picture of the combined behavior of several quantities, while exact counting results in combinatorics are usually capable of capturing the joint behavior of only two quantities.
Location: Richardson Building - RB-117 (G)
November 13
Title: Slack Matrices of Affine Semigroups
Speaker: Amy Wiebe - University of BC, Okanagan
Abstract: Slack matrices of polyhedral cones are an important class of nonnegative matrices. They offer canonical representations for cones that can be used for the study of realization spaces of polytopes and are a main ingredient in Yannakakis’ seminal result on lifts of polyhedral cones.
In this talk we generalize the notion of slack matrices to affine semigroups - a discrete analog of polyhedral cones - and present the corresponding result relating lifts of affine semigroups to nonnegative integer factorizations of their slack matrices. We use these generalizations to present new results on nonnegative integer rank of integer matrices.
Location: Richardson Building - RB-117 (G)
Time: 3:00
November 20
Title: Rees algebras of linearly presented ideals
Speaker: Alessandra Costantini - Tulane University
Abstract: Rees algebras represent an essential algebraic tool in the study of singularities of algebraic varieties, as they arise, for instance, as homogeneous coordinate rings of blowups or graphs of rational maps.
In this talk, I will discuss the problem of finding the defining equations of Rees algebras. Although this is wide open in general, the problem becomes treatable in the case of height-two perfect ideals with a linear presentation, where one can use a combination of homological methods and linear algebra, inspired by classical elimination theory.
This is part of joint work with E. Price and M. Weaver (arxiv:2308.16010 and arxiv:2409.14238
Location: Richardson Building - RB-117 (G)
Time: 3:00
