Jordan Ellenberg (University of Wisconsin, Madison) Congruence subgroups of mapping class groups (especially braid groups.) Abstract: Abstract: There is a notion of "congruence subgroup" of a mapping class group which appears to have certain properties in common with congruence subgroups of arithmetic groups. We will discuss several open problems in the theory of congruence subgroups, some progress, and applications: for instance, the proof of a "strong congruence subgroup property" for the mapping class group of a oncepunctured elliptic curve answers an open question from dynamics about Teichmuller curves in moduli spaces of curves (joint work with D.B. McReynolds). In number theory, theorems about stabilization of cohomology of congruence subgroups yield results towards the CohenLenstra heuristics for ideal class groups over function fields over finite fields. (joint work with A. Venkatesh, C. Westerland.) 
