PUNDiT: Practicum for Undergraduates in Number Theory

October 21 - 22, 2023

Expository Talks

Title: Fun Times with Algebraic Groups

Name: Cris Negron

Abstract: I will talk about algebraic groups; who they are, and what they do. Algebraic groups provide a beautiful meeting place where concepts from algebra, geometry, representation theory, and even mathematical physics come together to help us organize our lives as mathematicians. I will touch on the classification of simple algebraic groups via low-level combinatorial objects called Dynkin diagrams, and if time permits we’ll conclude with a glimpse into the world of quantum groups.


Title: Chromatic Polynomials

Name: Rosa Orellana

Abstract: A graph consists of vertices and edges specifying a relation between any two vertices. A simple graph allows only one edge between vertices. Coloring the vertices has applications to scheduling problems. The chromatic polynomial of a graph is a one variable polynomial which encodes the number of ways to color a graph with a given number of colors. In the mid-1990s Stanley introduced a multivariable polynomial that generalizes the classic one variable chromatic polynomial. What properties of a graph can be recovered from this polynomial? A tree is a connected graph without cycles. It is conjectured that Stanley’s polynomial distinguishes trees up to isomorphism. In this talk, I will discuss some recent results on this conjecture, some obtained by undergraduate students.


Title: Introduction to the Hasse Principle

Name: Lori Watson

Abstract: The hope behind the Hasse Principle is that one can determine whether a Diophantine equation is solvable by determining whether the equation is solvable modulo every positive integer n. In this talk, we will discuss some of the motivation behind the Hasse Principle, its usefulness in determining whether there are rational points on algebraic curves, and some of its limitations.


Title: The Geometry behind Fermat’s Last Theorem

Name: Anthony Várilly-Alvarado

Abstract: Fermat’s last theorem states that for all whole number $n \geq 3$, the only integer solutions to the equation $x^n + y^n = z^n$ must have at least one of $x$, $y$, or $z$ being zero. Fermat claimed to have a proof of this theorem in the 17th Century, but there is no evidence he did. Andrew Wiles, aided by Richard Taylor, proved Fermat’s Last Theorem in 1995. Well before Wiles’ proof, there were strong reasons to believe the theorem was true, coming from a different part of Mathematics: geometry. In this talk I will explain the geometry behind Fermat’s equation