M@X — Math @ Xavier

Algebraic Statistics is an emerging field of research that uses techniques from Algebraic Geometry, Combinatorics and Commutative Algebra to enhance our understanding of statistical and causal inference problems. A key area of research in this field is graphical models, where the dependence structure between random variables is determined by a graph. In this talk, I will define the key concepts of Gaussian graphical models, conditional independence and its connection with algebra, which will be accessible to undergraduate students.

This will be an introduction to my favorite area of algebraic research. I intend to make the presentation broadly accessible, focusing on the larger picture rather than on specific technicalities. Hopefully, the talk will arouse some interest among listeners to potentially dive deeper into some of the topics I will mention.

Determining the automorphism group of an algebra is, in general, a very difficult problem. However, certain types of noncommutative algebras have relatively few automorphisms, and in a few cases, the automorphism group can be explicitly described. In this talk, I will discuss our recent work on the automorphism problem for quantum Schubert cell algebras, including recent developments and some open problems. This is joint work with H. Melikyan.

We consider systems consisting of one conservation law coupled with a balance law, as well as systems of two balance laws, and investigate the structure of Riemann solutions arising from piecewise constant initial data. While classical theory describes solutions in terms of shocks, rarefactions, and contact discontinuities, specific systems admit nonclassical behaviors, including singular solutions such as overcompressive singular or delta shocks. These arise in models often motivated by physical phenomena in gas dynamics, traffic flow, and biological aggregation, among others. In this talk, I will present a range of nonclassical solution structures that emerge in such systems, highlighting scenarios where classical entropy conditions fail to select a unique solution. We discuss analytical techniques used to construct and classify these solutions, including the use of regularization methods such as Dafermos’ viscous approximation and geometric singular perturbation theory. The theoretical findings are numerically confirmed using the Local Lax-Friedrichs scheme.

In 1914, Hardy proved the existence of infinitely many non-trivial zeros of the Riemann zeta function on the critical line Re(s)=1/2 using the Jacobi theta relation. In this talk, we shall first discuss a number field analogue of the Jacobi theta relation and, as an application, show the existence of infinitely many non-trivial zeros of the Dedekind zeta function on Re(s)=1/2. This is joint work with Diksha Rani Bansal.

This presentation is intended for undergraduate students studying mathematics. It is an advertisement for complex analysis. We will start with the remarkable Stirling’s formula which describes the behavior of N! When N is large, as a tool to introduce the asymptotic analysis of Riemann-Hilbert problems. Then we will delve into the connection to partial differential equations. There will be examples. Oh yes! There will be examples.