TATERS

Helly’s theorem from 1912 asserts that for a finite family of convex sets in a d-dimensional Euclidean space, if every d+1 of the sets have a point in common then all of the sets have a point in common. This theorem found applications in many areas of mathematics and led to numerous generalizations. Its proofs are very elementary and are suitable for undergraduate students and advanced high-school students.  

Helly’s theorem is closely related to two other fundamental theorems in convexity: Radon’s theorem asserts that a set of d + 2 points in d-dimensional real space can be divided into two disjoint sets whose convex hulls have non empty intersection. Caratheodory’s theorem asserts that if S is a set in d- dimensional real space and x belongs to its convex hull then x already belongs to the convex hull of at most d + 1 points in S.

We will discuss several developments around Helly’s theorem. 

One particular deep and mysterious theorem in this area is Tverberg's theorem that asserts that n points in d-dimensional Euclidean space with n = (d+1)(r-1)+1, can be divided into r parts whose convex hulls have non-empty intersection. 

If time allows we will discuss various quantitative versions of Helly’s theorem, "(p, q)"- theorems, and the beautiful Amenta’s theorem (a result about families of unions of convex sets). We will also mention some connections to topology.

What does prime characteristic have that characteristic zero doesn't? For one thing, the ``Freshman's dream'' holds: if I have polynomials with coefficients in $\mathbb F_2$, then $(f(x)+g(x))^2=f^2(x)+g^2(x)$. This seemingly simple property gives rise to a surprisingly powerful tool, namely, the Frobenius map. We'll look at how algebraic properties of the Frobenius can be used to study geometric properties of varieties. We'll also see how questions in complex algebraic geometry can sometimes be reduced to questions in prime characteristic, and some connections between the geometry in these two settings.

Lefschetz properties of apolar algebras are modeled on the Hard Lefschetz theorem for cohomology rings of complex projective manifolds. A substantial literature concerns determining which algebras have a Lefschetz property. I will introduce the Weak and Strong Lefschetz properties, briefly describe new work showing that the apolar algebras of the generic determinant and permanent have the Strong Lefschetz Property, and discuss how one might apply Lefschetz properties in apolar algebras, particularly to questions about Waring rank.

A key branch of commutative algebra, combinatorial commutative algebra, focuses on the study of square-free monomial ideals using combinatorial structures. Many techniques have been developed to analyze these ideals, particularly through the use of simplicial complexes and (hyper)graphs. This talk explores square-free monomial ideals using finite simple graphs and their minimal totally dominating sets. We will examine how the algebraic properties of the ideal translate into graph-theoretic properties of its associated graph, highlighting the connection between these two fields. 

In a common NFL pool format, participants pick one team to win each week, with the restriction that each team can only be chosen once. I will show how combinatorial optimization can be used to maximize the expected number of correct picks in such pools and their variants. The talk will examine the computational complexity of these problems and their connections to matrix and tensor structures.

Polynomial functors are, essentially, polynomials whose variables take values in sets (or groupoids, etc.) rather than in numbers. In this talk, we will explore two related aspects of polynomial functors: first, polynomial functors provide a foundation for inductively defined data types; second, suitably finite polynomial functors yield categorified generating functions. We will look at these aspects of polynomial functors with binary trees as a recurring example, toward laying down the necessary machinery for outlining a sketch of a Catalan bijection involving a computational interpretation of derivatives of polynomial functors.

In modern physics, all physical interactions propagate through fields permeating spacetime.  Even before quantizing these fields, the classical theory of these fields has a rich mathematical structure.  Two key initial observations are: (1) the dynamics of these fields is governed by interesting partial differential equations; and (2) these equations exhibit symmetries which do not come from the symmetries of spacetime.  Gauge theory is the study of fields which exhibit such symmetries. (The term “gauge” in physics is just a choice of trivialization of some bundle.)

In this talk, I will give a tour of the history of gauge theory, a subbranch of differential geometry culminating in a breakthrough theorem in 4-manifold topology in 1983.  The first gauge theory was Maxwell’s theory of electromagnetism. The physical observables of this theory are the electric and magnetic fields, described in terms of auxiliary electric potentials and magnetic potentials which are required to satisfy a hyperbolic partial differential equation (in a vacuum, this PDE is simply the wave equation). We will discuss Maxwell’s equations, the Yang-Mills equations, the anti-self-dual Yang-Mills equations, culminating in Donaldson’s beautiful gauge-theoretic proof that not all topological 4-manifolds have a smooth structure.  

I will conclude by briefly talking about my research on the asymptotic geometry of the Hitchin moduli space.

Prerequisites: One of the joys and challenges of gauge theory is its interdisciplinary nature.  Though I will introduce it, prior exposure to manifolds and vector bundles would certainly be helpful.