Richard E. Phillips Lecture Series
January 9th-11th 2012
Peter Sarnak
Eugene Higgins Professor of Mathematics, Princeton University
Professor, Institute for Advanced Study
This spring's Phillips Lecture Series will be given by Professor Peter Sarnak during the second week of January, 2012 Please note times and locations below.
Peter Sarnak has made major contributions to number theory and to questions in analysis motivated by number theory. His interest in mathematics is wide-ranging, and his research focuses on the theory of zeta functions and automorphic forms with applications to number theory, combinatorics, and mathematical physics.
The following is the schedule of his lectures with abstracts:
- "Mobius Randomness and Dynamics" - Monday, January 9th, 4:10 - 5:00 p.m. Room:
115 International Center
Abstract: The Mobius Function mu(n) is minus one to the number of prime factors of n, if n has no square factors and zero otherwise. Understanding the randomness (often referred as the Mobius randomness principle) in this function is a fundamental and very difficult problem.We will explain a precise dynamical formulation of this randomness principle and report on recent advances in establishing it and its applications.
***There will be a Reception in the Mathematics Library (D101 Wells) immediately following this first talk. Everyone who attends the talk is invited.***
- "Thin Groups and the Affine Sieve" - Tuesday, January 10th, 4:10 - 5:00 p.m.
Room: A304 Wells Hall
Abstract: Infinite index subgroups of integer matrix groups like SL(n,Z) which are Zariski dense in SL(n), arise in geometric diophantine problems (eg Integral Apollonian packings) as well as monodromy groups associated with families of algebraic varieties. One of the key features needed when applying such groups in number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments connected with the affine sieve.
- "Nodal Lives of Maass Forms and Critical Percolation" - Wednesday, January 11th,
9:10 - 10:00 a.m. Room: A304 Wells Hall
Abstract: We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law. While proving appears to be very difficult, some approximations to it can be proved by developing various number theoretic and analytic methods. The work we report on is joint with A.Ghosh and A.Reznikov.