Kalina's personal website

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Richardson 108

Organizers: Alessandra Constantini and Kalina Mincheva.

September 3

Kalani Thalagoda, Tulane University.
A summation formula for mock modular forms. 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

Catherine Babecki.
Oriented matroids from non-polyhedral cones. 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

Shahriyar Roshan Zamir, Tulane University.
Interpolation in weighted projective spaces. 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 an example of a weighted projective plane, 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

Stephen Landsittel, Harvard University and Hebrew University of Jerusalem.
Generalized Hilbert Kunz Multiplicities of Families of Ideals. 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 1

Madeline Brandt, Vanderbilt University.
The weight-0 compactly supported Euler characteristic of moduli spaces of marked hyperelliptic curves. 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 6 - note the special date and place (Hebert 201)!

Jonathan Montano, ASU.
Homological ubiquity of quasi-poynomials in multigraded algebra. 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

Vivek Bhabani Lama, IIT Kharagpur.
Random Graph Functionals and Homological Invariants. 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

October 29

November 5

November 12

November 19

December 2