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:  Multiplicities and degree functions in local rings via intersection products

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

Abstract: We explore connections between intersection theory and multiplicity theory over Noetherian local rings. We begin by constructing an intersection product for schemes that are proper and birational over a Noetherian local ring, using the theory of rational equivalence developed by Thorup and the Snapper-Mumford-Kleiman intersection theory. This yields a new proof of a classical theorem of Rees on degree functions and leads to a generalization of Ramanujam’s formula for Hilbert-Samuel multiplicities to arbitrary Noetherian local rings. We also examine multiplicities and degree functions associated to graded families of m-primary ideals, especially divisorial filtrations in dimension two. This is joint work with Steven Dale Cutkosky.