Number Theory over Function Fields - Lecture 3

Explore the fascinating world of number theory over function fields in this comprehensive lecture by Will Sawin from Columbia University. Delve into the deep analogy between ordinary integers and polynomials in one variable over finite fields, as well as between number fields and function fields of algebraic curves over finite fields. Discover how applying geometric techniques to classical number theory problems involving polynomials over finite fields leads to new insights. Learn about recent progress in this field, including connections to other areas of mathematics such as the circle method for counting Diophantine equation solutions and its application to the topology of moduli spaces of curves in varieties. Examine geometric approaches to Cohen-Lenstra heuristics and their generalizations, which inspire new probabilistic results. Investigate the relationship between the analytic theory of automorphic forms over function fields and geometric Langlands theory. This two-hour lecture offers a rich survey of cutting-edge research in number theory, highlighting the power of geometric perspectives in solving classical problems.