# LSU Mathematics Student Colloquium

## Student Colloquium Menu

The goal of the LSU Mathematics Student Colloquium is to give both undergraduate and graduate students the opportunity to hear and interact with speakers from across the country, providing information and perspective possibly relevant to their graduate and postgraduate careers.

Each invited speaker will spend several days at LSU, giving multiple talks and making himself or herself available to undergraduates.

Talks are not confined to the math department but open to everyone. Those majoring in related fields are encouraged to come.

We are a chartered LSU student organization (constitution and bylaws). We are munificently funded by the Student Government Programming, Support, and Initiatives Fund (PSIF), and the LSU Mathematics Department. We are grateful for the generous past funding made by grants from the National Science Foundation (a VIGRE grant) and the Board of Regents.

## Fall 2022

### October 31-November 1, 2022: Henry (Hal) Schenck

Hal is a Professor and Rosemary Kopel Brown Eminent Scholars Chair in the Mathematics Department at Auburn University. His research interests are interdisciplinary: his background is in commutative algebra and algebraic geometry. He is especially interested in problems which can be studied from a computational standpoint, and in interactions with problems in applied mathematics.

**Undergraduate Talk** ( Lockett 232; Monday, October 31, 3:30- 4:30 PM)

Title: Combinatorics and Commutative Algebra

Abstract: This talk will give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and objects from algebraic geometry (toric varieties). No prior knowledge of any of the terms above will be assumed or needed for the talk.

**Graduate Talk** ( Lockett 276 ; Tuesday, November 1, 3:30-4:30 PM)

Title: Numerical Analysis meets Topology

Abstract: One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM). A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, there are many open questions about splines: for a triangular mesh $T$ and smoothness order one, the dimension of the vector space $C^1_3(T)$ of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a. formula for the dimension of the space $C^1_2(T)$ in terms of the combinatorics and geometry of the mesh $T$, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology, as well as interesting open problems.