Kalina's personal website

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.

October 2

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.

October 9

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.

October 16

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.

October 23

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.

October 30

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.

November 6

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.

November 13

November 20

December 4