Fall 2025

The Escape Rate for Polynomials

Speaker:  Ilaria Seidel

Abstract:  In this talk, we explore a bridge between complex dynamics and potential theory. Given a polynomial f, how fast do points escape to infinity under iterated applications of f? We first show that the escape rate is given by the Green’s function for the complement of the filled Julia set. We then give a dynamical description of the unique energy-minimizing measure on the Julia set and, time permitting, discuss why it coincides with the unique entropy-maximizing measure.

The Hanging Problem: A group fundamental to art

Speaker: Josh Rooney

Abstract: The year is 1895. A wealthy art collector is sick of looking at all her paintings and wants to donate them to Henri Poincaré. In her annoyance at their beauty, she donates them with one stipulation: each one must be hung from a singular string wrapped around two nails such that if either nail is removed, the painting shall fall. In this talk, we will study Poincaré's brilliant hanging technique. Whether this problem looks like a math problem or a riddle to you, homotopy theory awaits. This talk is intended to be exceptionally approachable to all undergraduates regardless of academic discipline.

Large Cardinals, Left-Distributivity, and Laver Tables

Speaker: Preston Bushnell

Abstract: A large cardinal axiom is roughly a set-theoretic statement asserting that the universe of sets is large. A left-distributive algebra is a set $X$ together with a binary operation $*: X \to X$ set $A$ together with a binary operation $*: A \times A \to A$ satisfying $a * (b * c) = (a * b) * (a * c)$ for all $a, b, c \in A$. This talk is about a series of discoveries from the 1990s which connected the two concepts: one large cardinal axiom, called \textit{Axiom I3}, gives rise to a left-distributive algebra $\mathcal A_j$. While $\mathcal A_j$ arises most naturally in set theory, it can also be expressed in terms of \textit{Laver tables}, a sequence of finite algebras that are definable in arithmetic. However, some of the properties of the Laver tables seem to require large cardinals to be proven. The question of how strong a theory is needed to prove these properties is deeply unsolved, and the gap between the upper and lower bounds could hardly be larger. We give a gentle introduction to all of this open problem and all of the concepts that define it. This talk is based on thesis research advised by Peter Koellner and W. Hugh Woodin.

The Eckmann-Hilton Argument

Speaker: Vincent Costa

Abstract: The Eckmann-Hilton argument is a result about how the "compatibility" of two unital multiplication structures implies that they are the same, and in particular, that they are commutative. In this talk, we will state and prove the main theorem, discuss a more abstract formulation in terms of monoid objects and monoidal categories, and use it to prove that higher homotopy groups are abelian.

Collusion in Auctions

Speaker: Jinho Park

Abstract: The talk will cover some background material about some basic auction types, their bidding properties in the symmetric IPV case, and then a walkthrough of a motivating example for the main results in his thesis.

Graduate Student Panel

Segments and Convexity in Metric Spaces

Speaker: Tian Vlasic

Abstract:  This talk is based on my work that is currently under submission to the Rose-Hulman Undergraduate Mathematics Journal. We are usually interested in studying convex sets in vector spaces. However, relevant research has shown that a meaningful notion of convex sets can be introduced to metric spaces, based on a more fundamental notion of a metric segment. In this talk, we will explore the properties of metric segments and convex sets in metric spaces. We will discuss the main theorems of my paper that indicate a link between the notions of a U-, directed, Menger convex, and strictly convex metric space. Finally, we will conclude the talk by exploring how convex sets in a metric space interact with the inherent topology of the metric space.