2010-2011 Milliman Lectures
Department of Mathematics
University of Washington

RICHARD TAYLOR
Herchel Smith Professor of Mathematics
Harvard University

March 1st, 2nd, and 3rd
4:00 - 5:00pm
Smith Hall, Room 120

Lecture I:
Reciprocity Laws and Density Theorems
(Tuesday, March 1st)

Abstract: If one fixes a polynomial (or a system of polynomials) in one or more variables, one can ask how the number of solutions modulo a prime number p varies with the prime p. Reciprocity laws give a formula involving completely different areas of mathematics (discrete subgroups of Lie groups). Density theorems give statistical information on how the number of solutions varies with p.

In the first lecture I will present a leisurely historical introduction to reciprocity laws and density theorems, starting with Gauss' law of quadratic reciprocity and finishing with the Sato-Tate conjecture. This lecture will also sketch Serre's approach to deducing density theorems from reciprocity laws.
View video of the lecture here.

Lecture II:
Galois Representations
(Wednesday, March 2nd)

Abstract: In the second lecture I will talk about a more general framework for discussing reciprocity laws. I will introduce Galois representations, L-functions and automorphic forms, and discuss their relevance in number theory. I will describe Langlands' very general reciprocity conjecture and the Fontaine-Mazur conjecture.
View video of the lecture here.

Lecture III:
Automorphy
(Thursday, March 3rd)

Abstract: I will describe what we currently know about general reciprocity theorems. I will give some indication of the techniques we have and what I see as the main stumbling blocks to further progress.
View video of the lecture here.

Bio: Richard Taylor received his B.A. from Clare College, Cambridge, and his Ph.D. from Princeton in 1988. He taught at Cambridge for six years before holding the Savilian Chair of Geometry at Oxford University 1995-1996. Since that time, he has taught at Harvard University and is currently the Herchel Smith Professor of Mathematics. For his work in number theory, he has been awarded the Whitehead Prize in 1990, the Fermat Prize in 2001, the Cole Prize in 2002, and the Shaw Prize in 2007.

Richard Taylor has been at the forefront of developments in number theory for the past twenty years. He collaborated with Andrew Wiles on their celebrated proof of Fermat's Last Theorem, and with Michael Harris on the proof of the local Langlands conjecture. He was awarded the Shaw Prize for his work on the Langlands program.