Lecture Series

Burgess’s method for short character sums: new directions

Lillian Pierce (Duke University and Bonn Research Fellow/HCM)


Date, Time, and Location:

Tuesday, June 25, 2019
Lecture I
11 am - 12 pm, Lipschitzsaal, Mathezentrum, Endenicher Allee 60, Bonn

Wednesday, June 26, 2019
Lecture II
11 am - 12 pm, SR 1.008, Mathezentrum, Endenicher Allee 60, Bonn

Thursday, June 27, 2019
Lecture III
12 pm - 1 pm, SR 1.008, Mathezentrum, Endenicher Allee 60, Bonn

Friday, June 28, 2019
Lecture IV
11 am - 12 pm, SR 0.008, Mathezentrum, Endenicher Allee 60, Bonn

 

Abstract

A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an additive or multiplicative character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve method, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly difficult to treat. In the colloquium for a general audience, we will see what makes a sum “short,” sketch why it would be powerful to understand short sums, and discuss Burgess's curious proof from the 1950’s for short multiplicative character sums. We will then describe new theorems, which push the Burgess method in new directions. In the subsequent series of three lectures, we will provide a deeper understanding of how Burgess’s original method works (in a streamlined modern presentation), and how certain new ideas are capitalizing on beautiful connections to other areas, such as harmonic analysis and algebraic geometry, in order to prove Burgess-type bounds in more general settings. 


The first lecture is colloquium-style for a general maths audience and the other three lectures are specifically addressed to local postdocs, PhD students, and master students. No registration needed.