Activities This Week

The theory of characters of infinite groups, initiated by Thoma, is a generalization of the representation theory of finite groups. More explicitly, a character of a group is an (extremal) conjugation invariant positive definite function. A group said to be character rigid if every character of the group is either supported on the center or comes from a finite dimensional representation. Connes conjecture that any irreducible lattice in a higher rank Lie group is character rigid. Surprisingly, this conjecture is a generalization of the celebrated Margulis normal subgroup theorem and of the Stuck-Zimmer conjecture on IRS rigidity. I will discuss a recent joint work with Alon Dogon, Yuval Gorfine, Liam Hanany, and Arie Levit showing that any nonuniform higher rank lattice is character rigid, proving the Stuck-Zimmer conjecture for such lattices.

$C^\ast$-covers of operator algebras, first studied by Arveson about 50 years ago, remain a vibrant area of research. Recently, Adam Humeniuk and Christopher Ramsey studied the lattice of $C^\ast$-covers of operator algebras, focusing on its structure and on whether this lattice uniquely determines an operator algebra up to completely isometric isomorphisms.

In my talk, I will give an introduction to operator algebras and their $C^\ast$-covers, and answer the question of whether there exists nontrivial operator algebras with a one point lattice. Additionally, I will characterize the possible cardinalities of the lattice of $C^\ast$-covers. This is a joint work with Adam Humeniuk and Christopher Ramsey.

Chromatic homotopy theory aims to study cohomology theories through a hierarchy of simpler layers, organized by a notion called height. In this talk I will introduce the basic ideas behind this viewpoint and explain two approaches to analyzing these monochromatic layers: the classical K(n)-local category, which is closely related to one-dimensional formal group laws, and the T(n)-local or telescopic category, which is more directly tied to periodic phenomena in the stable homotopy groups of spheres. I will then describe a framework for understanding periodicity inside the chromatic layers, and explain how this allows one to lift Picard elements from the K(n)-local setting to the telescopic setting. Finally, I will present an application to chromatic Galois theory, leading to the construction of a first example of a non-abelian Galois extension in the T(n)-local world.