Speaker
Takashi Shioya
(Tohoku)
Description
In this talk, I will discuss joint work with Takayuki Okuda (Hiroshima University) on the coarse geometry of metric measure spaces. Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called measured Block-Weinberger homology. We prove that this homology is invariant under measured coarse equivalence and, under the local doubling condition, that the vanishing of its 0-th homology is equivalent to non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures. Some examples will also be presented.