Delocalized eta invariants for self-adjoint first order elliptic differential operators on closed manifolds were first introduced by Lott, as a natural extension of the classical eta invariant of Atiyah-Patodi-Singer. It is a fundamental invariant in the studies of higher index theory on manifolds with boundary, positive scalar curvature metrics on spin manfolds and rigidity problems in topology. The delocalized eta invariant, despite being defined in terms of an explicit integral formula, is difficult to compute in general, due to its non-local nature. In this talk, I will report on some recent results concerning when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces, where the latter are easier to compute. This talk is based on joint work with Jinmin Wang and Guoliang Yu.
Abstract

The rational strong Novikov conjecture is a prominent problem in noncommutative geometry. It implies deep conjectures in topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that this conjecture holds for any discrete group admitting an isometric and proper action on a (possibly infinite-dimensional) nonpositively curved space that we call an admissible Hilbert-Hadamard space, partially extending earlier results of Kasparov and Higson-Kasparov. In particular, our result can be applied to geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold, as they act on an infinite-dimensional symmetric space called the space of L^2-Riemannian metrics. A crucial ingredient of our proof is the construction of C*-algebras from infinite dimensional nonpositively curved spaces. This is joint work with Sherry Gong and Guoliang Yu.
Abstract

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients, direct sums, and quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete. Joint work with Andrew Swift.