Number Theory over Function Fields

Explore the profound connections between classical number theory and algebraic geometry through this comprehensive lecture series delivered by Will Sawin from Columbia University at the Institut des Hautes Etudes Scientifiques. Delve into the deep analogy between ordinary integers and polynomials over finite fields, as well as the relationship between number fields and function fields of algebraic curves over finite fields, building upon foundational work by André Weil. Discover how geometric techniques unavailable in classical settings can be applied to polynomial analogues of traditional number theory problems, opening new avenues for mathematical investigation. Learn about recent progress in applying the circle method for counting Diophantine equation solutions to study the topology of moduli spaces of curves in varieties. Examine geometric approaches to the Cohen-Lenstra heuristics and their generalizations, understanding how they motivate purely probabilistic results. Investigate the connections between the analytic theory of automorphic forms over function fields and geometric Langlands theory. Gain insight into how the geometric perspective creates bridges to diverse mathematical areas, demonstrating the power of cross-disciplinary approaches in modern mathematics. The series spans six detailed lectures, each building upon previous concepts to provide a thorough survey of this active research area where number theory, algebraic geometry, and geometric representation theory intersect.