Number Theory over Function Fields - Lecture 4

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 and connections with other areas of mathematics. Learn about the circle method for counting solutions to Diophantine equations and its application to the topology of moduli spaces of curves in varieties. Investigate geometric approaches to Cohen-Lenstra heuristics and their generalizations, which inspire new probabilistic results. Examine the connection between the analytic theory of automorphic forms over function fields and geometric Langlands theory. Gain a deeper understanding of recent progress in this field and its implications for various branches of mathematics in this nearly two-hour lecture presented at the Institut des Hautes Etudes Scientifiques (IHES).