Analysis Seminar: "The Hochschild Cohomology of Roe Type Algebras"

Speaker: Mathew Lorentz, Michigan State University

Abstract: Many times in analysis we focus on the ``small scale" structure of a metric space, e.g., continuity, derivations, etc. However, to examine the ``large scale" structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X,d_X) is coarsely equivalent to (Y, d_Y) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely topological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules. 

Host: Walton Green