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Noncommutative Geometry Seminar

Fall 2017

Date:August 30, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Guihua Gong, University of Puerto Rico
Title: Invariant and classification of inductive limit C*-algebras with ideal property
Abstract:After the recent sucessful classification of unital simple separable nuclear C*-algebras of finite decomposition rank due to Gong-Lin-Niu and Elliott-Gong-Lin-Niu, it is becomes important to seek possible generalization of the classification to non simple C*-algebras. A C*algebras A is said to have ideal property if each ideal I of A is generated by the projections inside the ideal. The class of C*algebras with ideal property is a common generalization of the class of unital simple C*algebras and real rank zero C*-algebras. In this talk, we will give a classificsation of AH algebras (of no dimension growth) with ideal property. In this classification, it invloves a new ingrent in the invariant: compatibility of Hausdorffized algebraic K_1 group.

Date:September 6, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Shilin Yu, Texas A&M University
Title:Towards a geometric understanding of Mackey-Higson bijection
Abstract:Connes and Higson observed that the well-known Baum-Connes-Kasparov conjecture in operator algebra suggests a mysterious bijection between the tempered dual of a real reductive group and that of its Cartan motion group, which was already conjectured by Mackey in 1970's. In this talk, I will show that this bijection follows naturally from families of D-modules on the flag variety. I will also discuss its relationship with Kirillov's coadjoint orbit method if time allows.

Date:September 13, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Xiang Tang, Washington University at St. Louis
Title: A longitudinal index theorem on an open foliation manifold
Abstract:In this talk, we will introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we will define an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. And we will discuss applications of our developments to longitudinal elliptic operators on an open foliated manifold. This is joint work with Rufus Willett and Yi-Jun Yao.

Date:September 20, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Yi Wang, Texas A&M University
Title:On the p-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains
Abstract:We show that under a mild condition, a principal submodule of the Bergman module on a strongly pseudoconvex domain, generated by a holomorphic function defined on a neighborhood of its closure, is p essentially normal for p>n. Two main ideas are involved in the proof. The first is that a holomorphic function defined in a neighborhood 'grows like a polynomial'. This is illustrated in a key inequality that we prove in our paper. The second is that commutators of Toeplitz operators behave much better than the operator themselves.

Date:September 29, 2017
Time:2:00pm
Location:*BLOC 220*
Speaker:Sherry Gong, Massachusetts Institute of Technology
Title:Marked link invariants: Khovanov, instanton, and binary dihedral invariants for marked links
Abstract:We introduce a version of Khovanov homology for alternating links with marking data, $\omega$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology (Kronheimer and Mrowka, \textit{Khovanov homology is an unknot-detector}) collapses on the $E_2$ page for alternating links. We moreover show that the Khovanov homology we introduce for alternating links does not depend on $\omega$; thus, the instanton homology also does not depend on $\omega$ for alternating links. Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on $\omega$. (* Note the special time and room.)

Date:October 11, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Benben Liao, Texas A&M University
Title:Noncommutative maximal inequalities for group actions
Abstract:Let $G$ be a finitely generated group, and $M$ a semi-finite von Neumann algebra on which $G$ acts. When the group $G$ has polynomial growth, we obtain strong type $(p,p),p>1,$ and weak type $(1,1)$ maximal inequalities for $G$ acting on $M$. The result extends the work of Yeadon and Junge-Xu for $G$ being the integer group. This is based on joint work with Guixiang Hong and Simeng Wang (https://arxiv.org/abs/1705.04851).

Date:October 25, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Suleyman Kagan Samurkas, Texas A&M University
Title:Bounds for the rank of the finite part of operator $K$-Theory
Abstract:We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group $S(M)$ and the group of positive scalar curvature metrics $P(M)$ for an oriented manifold $M$. We define a class of groups called ``polynomially full groups'' for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator $K$-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.

Date:November 1, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Guoliang Yu, TAMU
Title:Higher eta invariants of elliptic operators and its applications
Abstract:We apply Suleyman Kagan Samurkas' recent result to extend John Lott's higher eta invariants to a more general setting. We prove a higher index theorem for manifolds with boundary using the higher eta invariants. We establish rationality of the higher eta invariants when the Baum-Connes conjecture holds. This is joint work with Zhizhang Xie.

Date:November 8, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Rufus Willett, University of Hawaii
Title:Finite dynamical complexity and controlled K-theory
Abstract:I’ll discuss a notion of finite dynamical complexity introduced in joint work in Erik Guentner and Guoliang Yu. This notion applies for topological dynamical systems (and more generally for étale groupoids). I’ll sketch connections to amenability and the earlier idea of finite decomposition complexity (introduced by Guentner, Tessera, and Yu), and applications to computing K-theory.

Date:November 30, 2017
Time:2:00pm
Location:*BLOC 220*
Speaker:Ronghui Ji, IUPUI
Title:From relative amenability to relative soficity for countable groups
Abstract:We define a relative soficity for a countable group with respect to a family of subgroups. A group is sofic if and only if it is relatively sofic with respect to the family consisting of only the trivial subgroup. When a group is relatively amenable with respect to a family of subgroups, then it is relatively sofic with respect to the family. We show that if a group is relatively sofic with respect to a family of sofic subgroups, then the group is sofic. This in particular generalizes a result of Elek and Szabo. An example of relatively amenable group G with respect to an infinite family of subgroups F is constructed so that G is not relatively amenable with respect to any finite subfamily of F.

Date:December 6, 2017
Time:2:00pm
Location:BLOC 628
Speaker:Xin Ma, TAMU
Title:Dynamics and classification of crossed product C*-algebras
Abstract:In this talk I will talk about some dynamical properties and their relation to classification program of crossed products by Elliott invariant. These properties include dimensions, almost finiteness, comparison, and the small boundary property. In addition, I will talk about some recent classification results for crossed products.