There is by now a broad body of work on minimal surfaces in positively curved ambient manifolds. If the ambient manifold has nonpositive curvature, much less is known. I will present some recent results on minimal submanifolds in nonpositively curved locally symmetric spaces, that are motivated by or have parallels to the positive curvature setting. The proofs bring new tools into the picture from representation theory. Another key ingredient is a new monotonicity formula for minimal submanifolds of low codimension in nonpositively curved symmetric spaces. 

I will then discuss applications to a program initiated by Gromov to prove statements of the following kind: Suppose we are given two manifolds X and Y, where X is “complicated” and Y is lower dimensional. Then any map f: X-> Y must have at least one “complicated” fiber. If time permits, I will also discuss some applications to systolic geometry, global fixed point statements for actions of lattices on contractible CAT(0) simplicial complexes, and/or non-abelian higher expansion and branched cover stability.  

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a “recipe” due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. We consider the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the moduli space of hyperelliptic curves, with symplectic coefficients. With Bergström-Diaconu-Westerland we compute these stable homology groups, together with their structure as Galois representations. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.