Lectures by Faculty

Primes and Equations

Richard Taylor
Institute for Advanced Study
February 1, 2012

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.

First Steps in Symplectic Dynamics

Helmut Hofer
Institute for Advanced Study
September 26, 2011

The modern theory of dynamical systems, as well as symplectic geometry, have their origin with Poincare as one field with integrated Ideas. Since then these fields developed quite independently. Given the progress in these fields one can make a good argument why the time is ripe to bring them closer together around the core area of Hamiltonian dynamics

Around the Davenport-Heilbronn Function

Enrico Bombieri
Institute for Advanced Study
November 10, 2011

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.

Education and Equality

Danielle Allen, UPS Foundation Professor
Institute for Advanced Study
November 16, 2011

Current educational policy discussions frequently invoke “equality” as the reigning ideal. But how clear a view do we have of what we mean by this? What exactly are we trying to achieve? In this lecture, Danielle Allen, UPS Foundation Professor in the School of Social Science, revisits the question of how to understand the ideal of equality in the context of educational policy.

Knots and Quantum Theory

Edward Witten, Charles Simonyi Professor
Institute for Advanced Study
October 28, 2011

A knot is more or less what you think it is—a tangled mess of string in ordinary three-dimensional space. In the twentieth century, mathematicians developed a rich and deep theory of knots. And surprisingly, as Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, explains in this lecture, it turned out that many of the most interesting ideas about knots have their roots in quantum physics.