Topics in analytic number theory
October - November 2024
Tim Browning, Stephanie Chan and Victor Wang
Synopsis
This course will have three parts:
- Part I: Counting rational points on algebraic varieties, by Browning
I will discuss various results, conjectures and techniques that have arisen through the
study of rational points of bounded height on varieties. I’ll briefly cover the Manin Conjecture, the dimension growth conjecture, the circle method, and some basic geometry of numbers techniques.
- Part II: Distributions in number theory, by Chan
I will provide a quick introduction to some classical topics in probabilistic number theory. I will discuss the connection between the distribution of additive arithmetic functions and independent random variables, and outline the proof of the Erdös-Kac Theorem using the method of moments.
- Part III: Algebraic structure and geometric estimates, by Wang
I will discuss classical work on point counting over finite fields, such as Dwork’s elementary proof of the rationality of zeta functions, or the Bombieri-Stepanov type elementary proofs of the Riemann hypothesis for curves. I will also discuss applications of finite fields and other algebraic structures, such as to recent joint work on matrix exponential sums or the nonabelian circle method.
Timetable
All talks are at 10:15-11:30 and take place in Mondi 3 with exceptions marked:
- Part I: Oct 8 (Lecture Hall), Oct 10 (Mondi 2), Oct 15, Oct 17
- Part II: Oct 22, Oct 24, Oct 29, Oct 31
- Part III: Nov 12, Nov 14, Nov 19, Nov 21
Course Assessment
The course will be assessed by problem sheets.
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