Noncommutative Geometry Seminar
Organizers:
Zhizhang Xie,
Guoliang Yu,
Jianchao Wu,
Hao Guo
Please feel free to contact any one of us, if you would like to give a talk at our seminar.

Date Time 
Location  Speaker 
Title – click for abstract 

05/27 2:00pm 
Zoom 942810031 
Zhizhang Xie Texas A&M University 
Approximations of delocalized eta invariants by their finite analogues
Delocalized eta invariants for selfadjoint first order elliptic differential operators on closed manifolds were first introduced by Lott, as a natural extension of the classical eta invariant of AtiyahPatodiSinger. 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 nonlocal 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 finitesheeted covering spaces, where the latter are easier to compute. This talk is based on joint work with Jinmin Wang and Guoliang Yu. Abstract 

06/03 1:00pm 
Zoom 942810031 
Paolo Antonini SISSA 
The Baum–Connes conjecture localised at the unit element of a discrete group
For a discrete group Γ we construct a Baum–Connes map localised at the group unit element. This is an assembly map in KK–theory with real coefficients leading to a form of the BaumConnes conjecture which is intermediate between the Baum–Connes conjecture and the Strong Novikov conjecture.
A second interesting feature of the localised assembly map is functoriality
with respect to group morphisms. We explain the construction and we show that the relation with the Novikov conjecture follows from a comparison at the level of KKRtheory of the classifying space for free and proper actions EΓ with the classifying space for proper actions EΓ.
Based on joint work with Sara Azzali and Georges Skandalis. Abstract 

06/10 1:00pm 
Zoom 942810031 
Jianchao Wu Texas A&M University 
The Novikov conjecture and C*algebras of infinite dimensional nonpositively curved spaces
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 GromovLawson conjecture on positive scalar curvature. Using C*algebraic and Ktheoretic tools, we prove that this conjecture holds for any discrete group admitting an isometric and proper action on a (possibly infinitedimensional) nonpositively curved space that we call an admissible HilbertHadamard space, partially extending earlier results of Kasparov and HigsonKasparov. 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 infinitedimensional symmetric space called the space of L^2Riemannian 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 

06/17 1:00pm 
Zoom 942810031 
Shintaro Nishikawa Penn State University 
Sp(n,1) admits a proper 1cocycle for a uniformly bounded representation
We show that the simple rank one Lie group Sp(n ,1) for any n admits a proper 1cocycle for a uniformly bounded Hilbert space representation: i.e. it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. Our construction is a simple modification of the one given by Pierre Julg but crucially uses results on uniformly bounded representations by Michael Cowling. An interesting new feature is that the properness of these cocycles follows from the noncontinuity of a critical case of the Sobolev embedding. This work is inspired from Pierre Julg's work on the BaumConnes conjecture for Sp(n,1). Abstract 

06/18 4:00pm 
Zoom 
Javier Alejandro ChavezDominguez University of Oklahoma 
Asymptotic dimension and coarse embeddings in the quantum setting
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 equicoarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a vertexisoperimetric inequality for quantum expanders, based upon a previously known edgeisoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete. Joint work with Andrew Swift. 

06/24 1:00pm 
Zoom 942810031 
Carla Farsi University of Colorado  Boulder 
Proper Lie Groupoids and their structures
I will talk about two projects in their final phase of completion.
(Joint with Scull and Watts) After defining the orbit category for transitive proper Lie groupoids and equivariant CWcomplexes, we define equivariant Bredon homology and cohomology theories for actions of transitive proper Lie groupoids by using similarities with the compact group action case. Our work can be seen as basic evidence for Morita equivalence invariance of general Bredon theories.
(Joint with Seaton) After defining groupoid Euler characteristics for cocompact proper Lie groupoids we prove that they can be realized as the usual Euler characteristic of groupoid inertias spaces. We prove that these Euler Characteristics are Morita invariant and extend those defined for orbifolds and Gspaces where G is a compact Lie group. Abstract 

07/08 1:00pm 
Zoom 942810031 
Aurélien Sagnier John Hopkins University 
TBA
