Arithmetic statistics for homotopy theorists, lecture 4

Lecture 4 by Ishan Levi.

Lecture series abstract: Arithmetic statistics aims to understand questions such as the distribution of class groups of a randomly chosen quadratic number field, or counting the asymptotic number of G-Galois extensions of bounded discriminant of the rational numbers. In this lecture series, I will explain how, in the function field setting, questions in arithmetic statistics may be reduced to questions about the homology of Hurwitz spaces, and how tools from homotopy theory can be used to answer those questions.

In the first lecture Ishan explained some of the basic questions in arithmetic statistics, such as the Cohen--Lenstra heuristics, Malle's conjecture and the Poonen--Rains heurstics over number fields, and indicated how these problems have analogous versions in the function field setting.

The second lecture, which can be understood independently of the first lecture,  will start by recalling Hurwitz spaces over function fields, and how statistical statements can be understood in terms of topology/homotopy theory. A key tool in this relation is the ability to reduce questions about understanding probability distributions to questions about understanding moments of the distributions, which will be explained in some detail.

Third+Fourth lecture: Cont.