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,dX) is coarsely equivalent to (Y, dY ) 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 bornological 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.
The Hochschild Cohomology of Roe Type Algebras
Thu, Apr. 13 3:35pm (MATH 3…
Tom Trogdon (University of Washington)
Many iterative algorithms from numerical linear algebra (Krylov subspace methods, in particular) connect directly to polynomials orthogonal with respect to a measure generated by a matrix and a vector, the so-called eigenvector empirical spectral distribution (VESD). For many random matrix distributions, the limiting VESD is known (or at least characterized) yet the ill-conditioned nature of the mapping from measure to orthogonal polynomials appears to limit the conclusions one can make. To partially overcome this difficulty, we develop a new Riemann--Hilbert-based perturbation theory for orthogonal polynomials that takes the local law as input. This allows one to make precise statements about the random polynomials as their degree diverges, and therefore, about algorithms applied to random matrices as the number of iterations diverges. This is joint work with Percy Deift (NYU), Xiucai Ding (UC Davis), and Elliot Paquette (McGill).
Local laws for random matrices, random orthogonal polynomials and algorithms