## EVENT DETAILS AND ABSTRACT

**Number Theory**
**Title:** Connectivity of Hurwitz spaces and the conjecture of Bourgain, Gamburd, and Sarnak

**Speaker:** Will Chen

**Speaker Info:** IAS

**Brief Description:**

**Special Note**:

**Abstract:**

A Hurwitz space is a moduli space of coverings of algebraic varieties. After fixing certain topological invariants, it is a classical problem to classify the connected components of the resulting moduli space. For example, the connectivity of the space of coverings of the projective line with simple branching and fixed degree led to the first proof of the irreducibility of M_g. In this talk I will explain a similar connectedness result, this time in the context of SL(2,p)-covers of elliptic curves, only branched above the origin. The connectedness result comes from combining asymptotic results of Bourgain, Gamburd, and Sarnak with a new combinatorial 'rigidity' coming from algebraic geometry. This rigidity result can also be viewed as a divisibility theorem on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. The connectedness is a key piece of information that unlocks a number of applications, including a conjecture of Bourgain, Gamburd and Sarnak on a strong approximation property of the Markoff equation x^2 + y^2 + z^2 - xyz = 0, a noncongruence analog of Rademacher's conjecture of the genus of modular curves, tamely ramified 3-point covers in characteristic p, and counting flat geodesics on a certain family of congruence modular curves.

**Date:** Friday, November 19, 2021

**Time:** 3:00PM

**Where:** Lunt 107

**Contact Person:** Ilya Khayutin

**Contact email:** khayutin@northwestern.edu

**Contact Phone:**

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