Craig Westerland

(Department of Mathematics, University of Minnesota)

Arithmetic Statistics Via Algebraic Topology

Date Friday, March 4, 2016

Various questions arising in arithmetic have probabilistic or statistical formulations: e.g., what is the probability that an integer is square-free? What proportion of imaginary quadratic number rings are unique factorization domains? What is the likelihood that my favorite finite group is the Galois group of a number field of degree n? Using the analogy between number fields and function fields, many of these questions can be recast in a new geometric setting, where tools from algebraic topology can be used to approach them. I'll focus on one particular question -- the Cohen-Lenstra heuristics on the distribution of class groups of imaginary quadratic number fields -- and work of Ellenberg, Venkatesh, and myself which addresses it.