A Lichnerowicz vanishing theorem for the maximal Roe algebra

Hao Guo  (Texas A&M University)

10:00-11:00, May 14, 2019   Science Building A510

Abstract:

It is a classical result of Lichnerowicz that the index of the Dirac operator on a closed, spin manifold vanishes in the presence of positive scalar curvature. When the manifold is non-compact but has bounded geometry, one can define a refinement of this index that lives in the K-theory of the maximal Roe algebra. The aim of this talk is to outline a proof of the fact that, if the manifold has uniformly positive scalar curvature, then this index vanishes. The argument relies on a subtle distinction between the usual Roe algebra and a uniform version of it; the latter allows one to avoid some of the analytical difficulties encountered when working with the former, in particular when trying to establish a version of the functional calculus in the maximal setting. This work is joint with Zhizhang Xie and Guoliang Yu.