# School of Mathematics

## 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.)

## Symplectic Dynamics Seminar: Symplectic Structures and Dynamics on Vortex Membranes

## CSDM: A Tutorial on the Likely Worst-Case Complexities of NP-Complete Problems

The P vs. NP problem has sometimes been unofficially paraphrased as asking whether

it is possible to improve on exhaustive search for such problems as Satisfiability, Clique,

Graph Coloring, etc. However, known algorithms for each of these problems indeed are

substantially better than exhaustive search, if still exponential. Furthermore, although a

polynomial-time algorithm for any one of these problems implies one for all of them, these

improved exponential algorithms are highly specific, and it is unclear what the limit of

improvement should be.

## Members Seminar: The Role of Symmetry in Phase Transitions

This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.

## Objectivity: The Limits of Scientific Sight

## Bilinearized Legendrian Contact Homology

## Around the Davenport-Heilbronn Function

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In this lecture, we study the Moebius inversion for functions of this type and show how its behavior is related to the distribution of zeros in the half-plane of absolute convergence. Work in collaboration with Amit Ghosh.