Mapping class groups and automorphism groups of free groups arise as the fundamental groups of the moduli spaces of surfaces and graphs. Finiteness properties of these groups, such as finite generation, are equivalent to finiteness of these moduli spaces. The Johnson filtration G(k) is (in both cases) a natural family of subgroups directly analogous to congruence subgroups of arithmetic groups, beginning with the famous Torelli group. The finiteness of these subgroups has been a major focus of low-dimensional topology for many years. I'll discuss a proof, joint with Mikhail Ershov and Andrew Putman, that G(k) is finitely generated for all k (as long as the genus/rank is large enough, say 4k). The main technical tool is the group-theoretic results of Bieri-Neumann-Strebel, which I'll explain from scratch (so anyone who knows the definition of a group should be able to get something out of this talk). This result was previously proved for k = 0 by Dehn (1922) and Nielsen (1924), for k = 1 by Magnus (1935) and Johnson (1983), and for k = 2 by Ershov--He (2017). Speaker(s): Tom Church (Stanford)