Gromov initiated a program to prove statements of the following form: Suppose we are given two simplicial complexes 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. In this talk I will describe various results of this kind for compact locally symmetric spaces, that are proved by bringing new tools into the picture from minimal surface theory and representation theory. Much of the talk will be focused on octonionic hyperbolic manifolds, the case where our approach seems to work best. If time permits I will also discuss some applications to systolic geometry, global fixed point theorems for actions of higher rank lattices on contractible CAT(0) simplicial complexes, and/or a connection to recent work by Connell-McReynolds-Wang involving homology vanishing theorems for infinite covolume discrete subgroups of SL(n,R). Based on joint work with Mikolaj Fraczyk.

In the first talk I will focus on giving background on Gromov's program for proving waist inequalities, and the necessary ingredients from representation theory and the theory of minimal submanifolds. The second talk will not assume that you went to the first talk.