Research Seminars: Algebra and Combinatorics

September 3

Title:  A summation formula for mock modular forms 

Speaker: Kalani Thalagoda - Tulane University

Abstract: Analytic number theorists frequently use summation formulas to study the asymptotic and statistical behavior of interesting (and sometimes erratic) arithmetic functions. For Dirichlet series satisfying a certain functional equation, Chandrasekharan and Narasimhan proved a formula for a weighted sum of the first n coefficients. In this talk, I will discuss a summation formula for mock modular forms of moderate growth and an application of it to Hurwitz class numbers. This is joint work with Olivia Beckwith, Nicholas Diamantis, Rajat Gupta, and Larry Rolen.

September 10

Title:  Oriented matroids from non-polyhedral cones

Speaker:  Catherine Babecki - California Institute of Technology Host: (Dan Bernstein)

Abstract: Existing generalizations of matroids to infinite settings are combinatorial in nature-- we propose a geometric alternative. One perspective on realizable oriented matroids comes from vector configurations and linear dependences among them. Pulling this back a step, the circuits (minimal dependences) are exactly the support-minimal vectors which lie in the null space of a linear map. We define conic matroids in a way that mimics this, and in particular, the "face-minimal" vectors in a subspace form a conic matroid analogously to standard realizable matroids. If the cone is the nonnegative orthant, we recover standard realizable oriented matroids. We will discuss our precise definitions, show how this structure captures features of Gale duality and conic programming, and share some of the directions we have yet to make headway in. Joint work with Isabelle Shankar and Amy Wiebe.

September 17

Title:  Interpolation in weighted projective space

Speaker: Shah Roshan Zamir - Tulane University

Abstract: Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992--1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this talk, we primarily use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space. We will also introduce an inductive procedure, originally due to A. Terracini from 1915, to demonstrate the only example of a weighted projective plane, of a particular family, where the analogue of the Alexander-Hirschowitz theorem holds without any exceptions. Furthermore, we will give interpolation bounds for an infinite family of weighted projective planes. There are no prerequisites for this talk besides some elementary knowledge of commutative algebra.

September 24

Title:  Generalized Hilbert Kunz Multiplicities of Families of Ideals

Speaker: Stephen Landsittel - Harvard University and Hebrew University of Jerusalem

Abstract: We discuss existence and volume equals multiplicity for generalized Hilbert Kunz Multiplicities for p-families of ideals. We also exhibit Minkowski inequalities for p-families.
 

October 01

Title:  The weight-0 compactly supported Euler characteristic of moduli spaces of marked hyperelliptic curves

Speaker: Madeline Brandt  -  Vanderbilt University

Abstract: Deligne connects the weight-zero compactly supported cohomology of a complex variety to the combinatorics of its compactifications. In this talk, we use this to study the moduli space of n-marked hyperelliptic curves. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for its weight-0 compactly supported Euler characteristic, as a virtual representation of S_n. This is joint work with Melody Chan and Siddarth Kannan.
 

October 06

Title:  Homological ubiquity of quasi-poynomials in multigraded algebra

Speaker: Jonathan Montañ0 - Arizona State University (Host): Alessandra Costantini

Abstract: In commutative algebra, functors such as local cohomology, Ext, and Tor applied to sequences of modules often grow quasi-polynomially, i.e., they grow periodically along finitely many polynomials. In this work we use the theory of tame modules from persistent homology and Presburger arithmetic to provide an explanation for this quasi-polynomial behavior in the multigraded setting. This is joint work with Hailong Dao, Ezra Miller, Christopher O’Neill, and Kevin Woods.
 

October 15

Title:  Random Graph Functionals and Homological Invariants

Speaker: Vivek Bhabani Lama  -  Affiliation: IIT Kharagpur  (Host): Tai Ha
 

Abstract: In this talk, we discuss the properties of homological invariants of random graphs under the Erdős–Rényi models. In particular, we focus on the law of large numbers for regularity, depth, v-numbers and other invariants of edge ideals, path ideals and cover ideals of Erdős–Rényi random graphs. (This is a joint work with Arindam Banerjee, Ritam Halder and Pritam Roy).
 

October 22

Title:  Degenerations of torus orbits and beyond

Speaker: Carl Lian - Washington University in St. Louis
 

Abstract: Let Gr(k,n) be the Grassmannian of k-planes in C^n. The standard torus action on C^n induces a torus action on Gr(k,n), whose orbits encode interesting combinatorial and geometric invariants. I will discuss explicit degenerations of the “generic” torus orbit closure into a union of Richardson varieties, and a further degeneration in the union of Schubert varieties. These give new proofs of cohomological formulas of Berget-Fink and Klyachko, respectively. I will also mention various extensions and open directions.
 

October 29

Title:  Hypergeometric Distributions and Joint Families of Elliptic Curves

Speaker: Hasan Saad - LSU 
 

Abstract: In the 1960’s, motivated by the Sato–Tate conjecture, Birch proved that the traces of Frobenius of elliptic curves over large finite fields is modeled by the semicircular distribution. Inspired by Birch’s result, in recent work, Ono, Saikia and Saad studied the distribution of finite field hypergeometric functions that relate to families of elliptic curves and K3 surfaces using modular forms and harmonic Maass forms. In this talk, we explore Sato–Tate type distributions for some finite field hypergeometric functions that relate to other families of curves and to joint families of elliptic curves in terms of the Meijer G-function. Moreover, we elucidate the appearance of Catalan numbers and Chebyshev polynomials in the earlier modular works.
 

November 05,

Title:  On the epsilon multiplicity and its density function

Speaker: Suprajo Das - IIT Madras  (Host): Tai Ha
 

Abstract: Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension $d$, and let $I\subseteq R$ be an ideal. Ulrich and Validashti defined the \emph{$\varepsilon$-multiplicity} of $I$ as
\[
\varepsilon(I) = \limsup_{n \to \infty}
\frac{\lambda_R\!\left(H^0_{\mathfrak{m}}(R/I^n)\right)}{n^d / d!}.
\]
This invariant may be viewed as a generalization of the classical Hilbert-Samuel multiplicity. Cutkosky showed that the $\limsup$ in this definition can be replaced by an actual limit when $R$ is analytically unramified. A surprising example due to Cutkosky, Hà, Srinivasan, and Theodorescu demonstrates
that this limit can be an irrational number even when $R$ is a regular local ring.

In this talk, we shall focus on the case of homogeneous ideals in a standard graded domain over a field. Motivated by Trivedi's approach to the Hilbert-Kunz multiplicity via density functions, we introduce a compactly supported continuous real-valued function, called the \emph{$\varepsilon$-density function}, whose integral recovers the $\varepsilon$-multiplicity. If time permits, we will present explicit examples and discuss applications in the context of integral closures.

This talk is based on joint work with Roy and Trivedi.
 

November 11,

Title:  Groupoids of Configurations of Lines

Speaker:  Jake Kettinger - Colorado State University
 

Abstract: In recent years, algebraic geometers began exploring a fascinating phenomenon exhibited by configurations of points in projective space wherein their general projection into a hyperplane is a complete intersection. This phenomenon was called the "geproci" property. In an effort to investigate this geproci property, Brian Harbourne and Allison Ganger began to look at special groups induced by configurations of lines known as groups of groupoids. In this talk, we will give some background on the geproci property and how it relates to configurations of lines known as spreads and maximal partial spreads. We will then define the groupoid induced by a configuration of lines in P^3 (with lots of Desmos demonstrations!), and look at specific examples of this groupoid.