Previous Talks

In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond. They were prominently featured in Wiles' proof of Fermat's last theorem, but there are many other problems that lead to a question about elliptic curves. We will discuss some of the open problems related to elliptic curves, including conjectures that are quite controversial! For a quick introduction to elliptic curves, see this Notices article (coauthored with Harris Daniels).

Since their discovery twenty years ago, cluster algebras have been widely studied within algebraic combinatorics, with deep connections to fields such as representation theory, total positivity, and theoretical physics.

In this talk, we will answer the question “What is a cluster algebra, and why is it a combinatorially compelling object?” To this end, we will explore the relationship between cluster algebras and two iconic combinatorial objects: Catalan numbers and a family of convex polytopes called associahedra. In particular, we will generalize both of these notions using exchange graphs of finite type cluster algebras.

This talk aims to shine a light on one of the fields of number theory: metric number theory. This is an expansive and active field of research with roots dating back at least to Dirichlet and his famous pigeonhole principle. While no talk could ever aim to provide an exhaustive cover of an entire field, this talk will introduce the fundamental problem: making measure theoretic statements about numbers or vectors with certain approximation properties. I will introduce the goal of Diophantine approximation and its connection to continued fractions. Then I will discuss improvements on Dirichlet's theorem, connections to ergodic theory, and talk briefly about the recently resolved Duffin-Shaeffer conjecture and the still open Littlewood's conjecture. This talk aims to be wide, not deep, so as to focus on how much room there is inside of metric number theory rather than on a specific problem.

Slides available here.

In this talk, we will introduce some finite incidence structures defined using geometric axioms; for example, projective planes, generalized polygons and polar spaces. These finite geometries have played an important role in various areas of mathematics. They have been used in the study of finite simple groups, linear codes, combinatorial designs, association schemes and extremal graphs. Recently, they have also been used with tremendous success in Ramsey theory. We will look at some of these interesting new applications.

Many objects in mathematics have a geometric analogue that can provide a different perspective on the same problem. Information geometry is the field of study where we consider information theory geometrically. We begin with some basic intuition and refreshers about probability, consider bits and pieces of algebraic geometry, differential geometry, and topology before discussing some of the exciting applications and research in the field.

Do you know what algorithm is deciding which piece you get next in a Tetris game? In this talk I will start by answering this question and then I will tell you about several different ways of sampling random polyominoes (polyominoes are like tetrominoes but with any desired amount of squares). We will also analyze how the topological and geometric properties of polyominoes change depending on the distribution that we choose to sample them.

Slides available here

BCK-algebras are the algebraic semantics of a non-classical logic having implication as its only logical connective, and commutative BCK-algebras are a well-behaved sub-class of BCK-algebras. While we will use a tiny bit of propositional logic to serve as motivation, this talk will mostly be algebraic in nature, focusing significantly on examples. From there, we will introduce the notion of ideals, define an associated topological space called the spectrum, and discuss some properties of spectra. This process is very similar to the construction of the spectrum of a commutative ring, and an analogy will be drawn between ring theory and BCK theory.