Everytopic Seminar

 October 17th, 2008, 1:40 pm – 3:00pm, Goldsmith 226

 

 

Title: Pretentious behavior in number theory

Speaker: Leo Goldmakher, University of Michigan (Ann Arbor)

 

Abstract:

A few years ago, Granville and Soundararajan introduced the  

notion of what they called "pretentious behavior" - they proved that  

the magnitude of a character sum (i.e. a sum of chi(n) over 0 < n < x,  

where chi is primitive Dirichlet character) is large for some x if and  

only if the character chi 'pretends' to be a primitive character of  

opposite parity and small conductor. This led to several breakthroughs  

in the study of character sums, including the first unconditional  

improvement of the Polya-Vinogradov inequality in almost 90 years.  

Since then they have continued to explore pretentious behavior in  

other contexts, perhaps the most notable application being to a key  

step ('weak subconvexity') in Holowinsky and Soundararajan's recent  

proof of the holomorphic analogue of the Quantum Unique Ergodicity  

conjecture. I will discuss pretentious behavior in number theory and

describe some applications of the theory.