Undergraduate students will give talks about their summer research or directed reading.

Euler discovered the zeta function. In a remarkable paper written in 1749, Euler calculates several values of the zeta function in regions where the series definition of the zeta function does not converge. We discuss Euler's proofs and relations to a solar eclipse on July 25, 1748 which Euler also wrote about.

The Weil Conjectures were influential proposals by Andre Weil concerning the generating functions derived from counting solutions to systems of polynomial equations over finite fields (algebraic varieties). They led to a successful decades-long program to prove them, including the development of the framework of modern algebraic geometry and number theory. In this talk, we'll present them in a historical context.

The representation theory of Lie groups and Lie algebras has played a significant role in quantum theory since the earliest days of the subject. I'll review this story and then explain some of the ways that "quantization" now appears in more recent approaches to representation theory.

Finite state automata and regular languages are basic notions of computer science. We'll explain how they naturally appear out of categories of cobordisms with defects in dimension one.

Last semester, I learned about how one can use elliptic curves with complex multiplication to compute the ray class fields of an imaginary quadratic number field. We will state this precisely and walk through the proof of the result.

We will introduce the fractional Laplace operator and present some PDE models involving nonlocal operators. We focus on a nonlocal model for minimal surfaces which is based on a notion of the perimeter of a set that takes into account long range interactions between points.

In thermodynamics, phase transitions are defined as abrupt macroscopic changes of behavior in the system as external parameters (such as temperature) are changed. In this talk, I will explain the probabilistic theory of phase transitions in statistical physics. I will also touch upon some of Hugo Duminil-Copin's contributions to this field for which he won the Fields Medal this year. No background in probability theory will be assumed.

This talk will aim to explain how linear ordinary differential equations link some objects arising from algebraic topology, complex analysis, and algebraic geometry. We will discuss how the monodromy of vector bundles equipped with a holomorphic connection can be used to obtain representations of the fundamental group of a Riemann surface.

In this talk I will aim to give a broad strokes introduction to the field of general relativity. I will start by introducing the fundamental mathematical objects in the theory and build to a discussion of the stability problem for black holes in general relativity.

Will a random walker ever return to its initial position? It depends on the model! In this talk we will develop the background needed to understand how random walks behave differently in different dimensions.