# School of Mathematics

## Mechanisms Leveraging Arbitrary Set-Theoretic Belief Hierarchies

## Categorical Langlands Correspondence in Positive Characteristic

## Conference on Graphs and Analysis

## CSDM: Finding Needles in Exponential Haystacks

## Symplectic Dynamics Seminar: How Large is the Shadow of a Symplectic Ball?

## Primes and Equations

One of the oldest subjects in mathematics is the study of Diophantine equations, i.e., the study of whole number (or fractional) solutions to polynomial equations. It remains one of the most active areas of mathematics today. Perhaps the most basic tool is the simple idea of “congruences,” particularly congruences modulo a prime number. In this talk, Richard Taylor, Professor in the School of Mathematics, introduces prime numbers and congruences and illustrates their connection to Diophantine equations. He also describes recent progress in this area, an application, and reciprocity laws, which lie at the heart of much recent progress on Diophantine equations, including Wiles’s proof of Fermat’s last theorem.

## CSDM: A Survey of Lower Bounds for the Resolution Proof System

## Symplectic Dynamics Seminar: On Conjugacy of Convex Billiards

On one side, we show that conjugacy of different domains can't be C^1 near the boundary. In particular, billiard maps of the circle and an ellipse are both analytically integrable, but not C^1 conjugate. On the other side, if conjugate near the boundary s smoother, then domains are the same up to isometry.

(This is joint work with A. Sorrentino.)