The 2023 Department of Mathematics Green Chair will be Dr. Kevin Knudson, chair of the Department of Mathematics at the University of Florida. While at TCU, Dr. Knudson will be giving multiple talks aimed at different audiences.
Graduate Student Talk
2:00 pm Tuesday, March 28, TUC 352
Morse Theory: Smooth and Discrete
This talk will serve as a gentle introduction to Morse theory on manifolds and the corresponding discrete version due to Forman. The only prerequisites are a good handle on multivariable calculus and some basic linear algebra. Knowledge of simplicial complexes is useful, but not necessary (I’ll define them during the talk).
4:30 pm Tuesday, March 28, BASS 1011
Five Platonic Friends
A Platonic Solid is a polyhedron all of whose sides are congruent equilateral polygons such that the same number of polygons is adjacent to each vertex. Euclid proved that there are exactly five such objects: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. During our time together I will show you how to make origami models of these objects and prove to you that these are the only five. This involves Euler’s polyhedron formula V-E+F=2, which we will also prove.
1:00 pm Wednesday, March 29, TUC 243
Algorithms in Discrete Morse Theory
Forman’s discrete Morse theory has proved to be an extremely useful tool in topology, combinatorics, homological algebra, and data analysis. In this talk I will present some algorithms for generating discrete Morse functions in various contexts. For example, given a sample from a smooth function on a manifold, can we construct a discrete Morse function whose gradient mirrors the behavior of the smooth one? Given a family of such things, can we build a coherent family of discrete Morse functions that tracks births and deaths of critical points correctly? If time permits I will also talk about a stratified version of these constructions.
7:00 pm Wednesday, March 29, SWR LH3
Mathematics of Gerrymandering
We’ve all seen pictures of bizarrely contorted Congressional districts and thought that surely something must be wrong. But how can we prove it? After giving a brief history of gerrymandering and the various strategies legislators employ I will discuss several mathematical tools that can help detect when a map has been drawn with nefarious goals. These include the efficiency gap, probabilistic methods, and compactness measures. The talk will be accessible to a general audience (that is, no super-advanced mathematical knowledge is required).