Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Richardson Building 117(G)

Organizers: Daniel Bernstein, Mahir Can, Tài Huy Hà, Kalina Mincheva.

Fall 2024

• Brian Grove, Louisiana State University.
  Explicit Hypergeometric Modularity and Applications. 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.

• Shlomo Gortler, Harvard University.
  Reconstructing configurations and graphs from unlabeled distance measurements. abstract.
  Place a configuration of n points (vertices) generically in $\mathbb{R}^d$. 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.

• Vinh Nguyen, University of Arkansas.
  Symbolic Powers of Matroids. 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.

• Vinh Pham, Tulane University.
  Newton non-degenerate ideals in regular local domains. 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.

• Hop D. Nguyen, Institute of Mathematics, Vietnam Academy of Science and Technology.
  Asymptotic regularity and depth of invariant chains of edge ideals. 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.

• Greg Blekherman, Georgia Tech.
  Tropicalization in Combinatorics. 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.

• Amy Wiebe, University of BC, Okanagan.
  Slack Matrices of Affine Semigroups. 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.

• Alessandra Costantini, Tulane University.
  Rees algebras of linearly presented ideals. 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 ).

• Sudipa Das, Arizona State University.
  Asymptotic Colengths for Families of Ideals. abstract.
  This talk focuses on the study of asymptotic colengths for families of m-primary ideals in a Noetherian local ring (R,m). We investigate various families, including weakly graded families, inverse graded families (in any characteristic), and weakly p-families and weakly inverse p-families (in positive characteristic). A new analytic approach will be presented to demonstrate the existence of these limits. Additionally, we will discuss Minkowski-type inequalities, positivity results, and volume-multiplicity relationships for these families. This research is based on joint work with Cheng Meng.

Past seminars.