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Explore the fascinating world of number theory over function fields in this comprehensive lecture. 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 classical number theory problems can be translated into analogues involving polynomials over finite fields, allowing for the application of new geometric techniques. Survey recent progress in this field, with a focus on how the geometric perspective creates connections to other areas of mathematics. Learn about the circle method for counting solutions to Diophantine equations and its application to studying the topology of moduli spaces of curves in varieties. Examine geometric approaches to Cohen-Lenstra heuristics and their generalizations, leading to new probabilistic results. Investigate the connection between the analytic theory of automorphic forms over function fields and geometric Langlands theory. Join Will Sawin from Columbia University for this insightful two-hour lecture at the Institut des Hautes Etudes Scientifiques (IHES), offering a comprehensive overview of recent advancements in number theory over function fields.
