Minimal submanifolds and waist inequalities for locally symmetric spaces (Ben Lowe)

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Background talk (11am): Waist inequalities, Property T, and Higher Expansion

Abstract: Suppose we are given two simplicial complexes $X$ and $Y$, where $X$ is "complicated” and $Y$ is lower dimensional than $X$. Then must a map $f: X \to Y$ have at least one "complicated” fiber? This talk will give an overview of a program Gromov initiated to prove quantitative statements of this kind, called waist inequalities. Both this talk and the next will feature examples where what "complicated" means can be either geometric (e.g. volume) or topological (e.g., a measure of the largeness of the fundamental group.) To give a sense for the theory I will first talk about what is known for concrete examples like the sphere and the torus, before moving towards the wilder setting of negative curvature. Along the way, I will as time permits describe connections to scalar curvature, (higher versions of) property $T$, systolic geometry, and various notions of higher expander simplicial complexes and manifolds originating in computer science. This talk will move slowly and not assume prior knowledge in this area.

Research talk (3:45pm): Minimal Submanifolds and Waist Inequalities for Locally Symmetric Spaces

Abstract: This talk will focus on the case of nonpositively curved locally symmetric spaces. In addition to being the most natural non-positively curved spaces to study from the perspective of differential geometry, they also have strong connections to geometric group theory, number theory, and algebraic geometry. I will describe recent joint work with Mikolaj Fraczyk that establishes a number of different kinds of higher expansion properties for families of manifolds in this setting by bringing new tools into the picture from representation theory and minimal surface theory. One goal will be to explain how knowledge of the unitary representations of a semisimple Lie group can be used to study the geometry of the associated locally symmetric spaces. On the minimal surface side, we establish new monotonicity formulas, or volume growth estimates, for minimal submanifolds of low-codimension in nonpositively curved symmetric spaces. I will explain how this can be played against the information coming from representation theory to prove waist inequalities. This talk may not move as slowly as the first talk but it will not assume prior knowledge in this area.