Braided Vector Spaces and Arithmetic Statistics Over Function Fields

Explore the unified framework of braided vector spaces for understanding arithmetic statistics over function fields in this advanced mathematics lecture. Delve into recent developments in arithmetic statistics over function fields over finite fields, examining key questions such as the distribution of the ell-primary part of class groups in random quadratic extensions and counting G-covers of projective lines with bounded discriminants. Learn how Cohen-Lenstra heuristics and Malle's conjecture can be understood through the lens of braid group cohomology with coefficients in braided vector spaces. Discover new results on prime discriminants developed in collaboration with Mark Shusterman, and gain insight into the many partially understood questions that remain in this rapidly evolving field. The presentation connects abstract algebraic structures with concrete arithmetic problems, demonstrating how braided vector spaces provide a powerful unifying perspective for diverse statistical questions in arithmetic geometry over finite fields.