UTRGV Algebra & Number Theory Seminar

The Algebra & Number Theory Seminar meets on Tuesdays or Fridays at 1:00pm in EMAGC 3.502 during Spring '25. Talks will be given in a hybrid format. If you would like to give a talk in the seminar please contact Debanjana Kundu or Luigi Ferraro.

debanjana.kundu[at]utrgv.edu

luigi.ferraro[at]utrgv.edu

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Souvik Dey

Charles University

Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications

The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this talk is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is well known that the complexity and curvature of the residue field characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection or regular local rings via injective curvature of the ring or that of the module of Kähler differentials.

January 28th at 1pm

Souvik's Website

Alexey Glazyrin

University of Texas Rio Grande Valley

Energy minimization problems

Given a continuous 2-point kernel $K$, the total energy of a point configuration is defined as the sum of values of $K$ over all pairs of points. Minimizing energy defined this way may be interpreted as finding the optimal distribution of particles under the interaction defined by the kernel $K$. Problems of this type have a rich history and arise in various areas of mathematics and science. We are particularly interested in a measure-theoretic extension of such problems. Often optimal configurations tend to cluster and optimal measures appear to be discrete. In this talk, we will cover this phenomenon in the case of $p$-frame potentials.

February 4th at 1pm

Alexey's Website

Andrew Alaniz

University of Texas Rio Grande Valley

Towards trace and determinant formulas for generalized cubic dissection matrices

Dissecting the $q$-series of a modular form is a common tool used to study its Fourier coefficients. In a previous work, Huber--Alaniz introduced generalized cubic dissection operators along with their matrix representations on certain spaces of modular forms. In this talk, we discuss progress towards a problem that extends a pair of conjectures posed by Huber--Alaniz via linear algebraic methods. We also discuss some combinatorial problems arising upon reducing the matrices modulo a prime $p$.

February 11th at 1pm

Pavel Snopov

University of Texas Rio Grande Valley

An Algebraic Perspective on Persistent Homology

Persistent homology has emerged as a fundamental tool in topological data analysis, providing a robust framework for capturing the shape of data. At its core, persistent homology is governed by the structure of persistence modules, which can be studied through an algebraic lens. In this talk, we will introduce the general setting of persistent homology and discuss the structure theorem for finitely generated 1-parameter persistence modules. Moving beyond the 1-parameter case, we will explore multiparameter persistence, where classification becomes significantly more complex. Time permitting, we will also touch on recent developments in the algebraic study of persistence modules.