Monday, February 24th, 2003
2:30pm - 3:3opm, in Science 2-065

Robert Seeley

UMass Boston

Zeta Functions in Geometry and Physics

Abstract: Riemann introduced a certain function of a complex variable s, defined as zeta(s) = sum of n^(-s) for n=1, 2, .... The sum converges only when Re(s) is > -1, but it has a meromorphic extension to the entire complex plane, with a simple pole at s=1. This Riemann zeta function plays a large role in number theory; it is at the heart of the Riemann hypothesis, the most famous unsolved problem in mathematics. Similar "zeta functions" are associated with the Dirac and Laplace operators of geometry and physics. They are originally defined for complex s in some half plane, but have meromorphic extensions to the whole plane. The poles and residues of these extensions have a variety of applications, such as a "local index theorem" in geometry and "zeta function regularization" in quantum field theory. The talk will discuss some of the applications, and some techniques used to analyze these functions. Some familiarity with functions of a complex variable will be useful, but we aim to make the talk widely accessible.