Non-Commutative Geometry and Higher Index Theory

Speakers (cont.)

Speakers

Jesus Sanchez (Washington University in St Louis): jesuss@wustl.edu

Sheagan John (Colorado University): sheagan.john@colorado.edu

Ahmad Reza Haj Saeedi Sadegh (Northeastern University): a.hajsaeedisadegh@northeastern.edu

Guoliang Yu (Texas A&M): guoliangyu@tamu.edu

Angel Roman (Washington University in St Louis): angelr@wustl.edu

Xiang Tang (Washington University in St Louis): xtang@wustl.edu

Robin Deeley (Colorado University): robin.deeley@gmail.com

Kun Wang (Texas A&M): kwang@tamu.edu

Xiaoyu Su (Texas A&M): xiaoyuegongzi@tamu.edu

Jintao Deng (University at Buffalo): jintaode@buffalo.edu

Qiaochu Ma (Penn State): qkm5040@psu.edu

Jinmin Wang (Texas A&M): jinmin@tamu.edu

Xiaoyu Su (Texas A&M)

Topological stable rank of F.D.C. action

Abstract: Topological stable rank was first introduced by M. Rieffel in 1982, it can be seen as dimension of C*-algebra which directly generalizes the classical concept of dimension for compact spaces. More interestingly, topological stable rank can be used to control the matrix size of the K-theory element, which makes K-theory in some sense computable. I will be talking about one approach to calculate the topological stable rank of finite dynamical complexity (F.D.C.) system and explain the idea behind this quantitative method that mimics the idea of quantitative K-theory, in which case the associated C*-algebra could be simple. This is based on joint work with Bogdan Nica and Guoliang Yu.

Jintao Deng (University at Buffalo)

The equivariant coarse Baum-Connes conjecture

Abstract: Let $G$ be a countable discrete group and $X$ be a metric space with bounded geometry. Assume that the group $G$ acts on the $X$ properly by isometries. The equivariant coarse Baum-Connes conjecture asserts that the assembly map from the equivariant K-homology to the K-theory of the equivariant Roe algebra is an isomorphism. The equivariant Roe algebra encodes the large-scale geometry of the metric space $X$ and the $G$-action. The equivariant higher index of an elliptic differential operator lies in the K-theory of this algebra. However, the computation of the K-theory of this algebra is difficult. The equivariant coarse Baum-Connes conjecture provides algorithm to this K-theory. In this talk I will talk about the result that the equivariant coarse Baum-Connes conjecture holds when the acting group is amenable and the quotient space $X/G$ admits a course embedding into Hilbert space. This is based on a joint work with Q. Wang and B. Fu.

Qiaochu Ma (Penn State)

Quantization commutes with reduction and K-homology.

Abstract: The “Quantization commutes with reduction”, or briefly, that “[Q,R] = 0”, first appeared as the famous Guillemin-Sternberg conjecture, which was proved by Meinrenken and Tian-Zhang using different approaches. In this talk, we present a "[Q,R]=0” type result for compact symplectic manifolds at the level of K-homology, using the analytic localization technique of Bismut-Lebeau/Tian-Zhang and the asymptotic morphism between C-star algebras introduced by Connes-Higson. This is joint work with Nigel Higson and Yiannis Loizides.

Jinmin Wang (Texas A&M)

Scalar curvature rigidity and Llraull's theorem

Abstract: Llraull's theorem yields that one cannot increase the scalar curvature and the metric of the standard sphere simultaneously. Gromov conjectures this scalar curvature rigidity for incomplete metrics on spheres with two antipital points removed, and more generally warped product metrics. In this talk, I will present our proof of Gromov's conjecture under an extra condition, and a counter example to Gromov's original statement. The main tool is the Dirac operator and a Poincare-type inequality. I will also give a brief introduction to the mu-bubble approach to this problem in dimension four. The talk is based on joint works with Simone Cecchini, Zhizhang Xie, and Bo Zhu.

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