Talk

Recently, Gurevitch and Howe have associated a notion of "rank" to representations of the symplectic group, and showed that "highest rank" representations satisfy a form of Howe duality over finite fields (c.f. talk on the 20th). Montealegre-Mora and me then realized that the rank-deficient reps occurring in this context can be characterized in terms of certain quantum error correcting codes. The purpose of this talk is to explain the background of this development, i.e. why physicists care about these objects in the first place.

This will be an experimental “Jazz session” where both of us will be explaining our recent results together.
We’ll also be encouraging audience participation and are aiming not only at 4-manifolds experts.

In topology, there are two ways to think about fiber bundles on connected spaces. Generally speaking, parallel transport along loops starting and ending at a basepoint gives rise to a map of groups from the loop space (i.e., the space of loops) to the group of automorphisms of the fiber. This map determines, and is determined by, the bundle. A bundle also determines, and is determined by, a map of spaces from its base to the classifying space of the automorphisms of its fibre. 

Thinking of fiber bundles in two ways establishes a duality between connected spaces and groups.

Joint with Cheyne Glass, Micah Miller, and Thomas Tradler.

The goal of the talk is to present the basics of persistent homology which is one of the primary tools in  applied algebraic topology and topological data analysis.
We hope to explain the mathematical ideas behind it, some of its current challenges and relations to sheaf theory.

Faltings' almost purity theorem asserts that, in p-adic geometry, things become simpler when passing to certain highly ramified, largely non-geometric, coverings. The coverings in question are somewhat akin to covering an interval by a Cantor set. I will show how embracing this idea has led us (with Dustin Clausen) to recast the basic notions of topology, leading to a better setup in which to do algebra with topological groups.

Video: 

20190807_scholze.mp4