The Noncommutative Geometry Seminar is part of the Center for Noncommutative Geometry and Operator Algebras in the Vanderbilt University Math Department. It meets from 4:10pm to 5:00pm on Tuesdays in SC 1432 (unless otherwise stated).
The Noncommutative Geometry Seminar is organized by Gennadi Kasparov, Zhizhang Xie and Guoliang Yu; please get in touch with one of us if you would like more information.
Upcoming events:
| Speaker |
Date |
Title and Abstract |
| Bai-Ling Wang (ANU, Australia) |
January 17th, 2012 |
Title: Geometry of
D-branes and twisted index theorem Abstract: In string theory D-branes were proposed as a mechanism for providing boundary conditions for the dynamics of open strings moving in space-time. As D-branes themselves can evolve over time one needs to study equivalence relations on the set of D-branes. An invariant of the equivalence class is the topological charge of the D-brane which should be thought of certain index of Atiyah-Singer type operators. In this talk, I will explain the geometry of space-times with a background "flux", and propose a mathematical definition of D-branes. As an application, we will study the Atiyah-Singer index theorem for a given twisting datum defined by the "flux". |
| January 24th, 2012 |
Canceled |
|
| Hang Wang (Tsinghua University, China) |
January 31th, 2012 |
Title: On L^2-index
theorems Abstract: I will give an introduction on the L^2-index theorem and review my thesis work on L^2-index formula on properly cocompact group actions. Then I will survey on the L^2-index theorems for some noncompact spaces in preparation for my future research. |
| Bai-Ling Wang (ANU, Australia) | February 7th, 2012 |
Title: Twisted index
theory for foliations Abstract: For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes-Skandalis longitudinal index theorem. |
| Zhizhang Xie (Vanderbilt) |
February 14th, 2012 |
Title: Higher index
theorems, diffeomorphisms and positive scalar curvature Abstract: This is first part of my talk on my recent joint work with Guoliang Yu on problems of positive scalar curvature on manifolds through noncommutative geometric methods. Suppose $M$ is a complete Riemannian manifold with positive scalar curvature toward infinity. Let $M_\Gamma$ be a Galois covering of $M$ by a discrete group $\Gamma$. Then one can define a higher index class $\ind(D_\Gamma)\in K_0(C_r^\ast(\Gamma))$ of the Dirac operator $D_\Gamma$ over $M_\Gamma$. Now suppose $M_0$ and $M_1$ are two manifolds with positive scalar curvature towards infinity and moreover their $\Gamma$-coverings coincide towards infinity. Let $M_2$ be the manifold obtained by gluing $M_0$ and $M_1$ along some ends. We prove the following relative higher index theorem: \[ \ind(D_2) = \ind(D_0) - \ind(D_1) \in K_0(C_r^\ast(\Gamma) ),\] where $D_i$ is the Dirac operator on $(M_i)_\Gamma$. (Click here to see the pdf version) |
| Zhizhang Xie (Vanderbilt) |
February 21th, 2012 |
Title: Higher index
theorems, diffeomorphisms and positive scalar curvature Abstract: This is the second part of my talk. Now let $X$ be closed manifold with Riemannian metric $g_0$ of positive scalar curvature and let $\Psi$ be a diffeomorphism $X \to X$. The induced metric $g_1 = \Psi^\ast g_0$ also has positive scalar curvature. Endow $X\times \mathbb R$ with the metric $g_t + (dt)^2$ where g_t = g_0 for t=<0, g_t = g_1 for t >=1, and any smooth homotopy from $g_0$ to $g_1$ for 0< t <1. Then $X\times \mathbb R$ has positive scalar curvature towards infinity. Consider a $\Gamma$-covering $X_\Gamma$ of $X$ and the Dirac operator $D_\Gamma$ over $X_\Gamma\times \mathbb R$. We use the relative higher index theorem above to show that the image of $\ind(D_\Gamma) $ inside $ K_0(C_r^\ast(\Gamma\rtimes_{\Psi} \mathbb Z) )$ coincides with $\ind(D_{X_\Psi}) \in K_0(C_r^\ast(\Gamma\rtimes_{\Psi}\mathbb Z))$, where $X_\Psi$ is the mapping cylinder of $X$ under $\Psi$ and $D_{X_\Psi}$ is the Dirac operator over $X_\Gamma\times \mathbb R$ as a $(\Gamma\rtimes_{\Psi} \mathbb Z)$-covering of $X_\Psi$. (Click here to see the pdf version) |
| Yves de Cornulier (University of Paris-South, Orsay, France) |
February 28th, 2012 |
Title: On the
topological full group of Cantor minimal systems Abstract: To any homeomorphism T of the Cantor set K, we can associate a discrete group, consisting of those homeomorphisms of K that are piecewise powers of T, called the topological full group of T. I'll survey a few recent results about full topological groups of Cantor minimal systems. |
| March 6th, 2012 |
No seminar
due to Spring break |
|
| March 13th, 2012 |
No seminar
due to Shanks Workshop: Geometry and Analysis of Large
Networks |
|
| Walther Paravicini (University of Münster, Germany) |
March 20th, 2012 |
Title: Equivariant
kk-theory for Banach algebras Abstract: The so-called Bost conjecture is the little, Banach algebraic sister of the famous Baum-Connes conjecture for group C*-algebras. It was introduced and put to use by Vincent Lafforgue in his thesis and proposes a method of calculating the topological K-theory of L^1-group algebras. In the form given by Lafforgue, it allows only C^*-algebras as coefficients, a fact that limits the possibilty of carrying methods to prove inheritance properties of the Baum-Connes conjecture over to the Bost conjecture. In my talk, I will present a new kk-theory for Banach algebras called kk^{ban}. It is defined in terms of universal constructions in a way that ensures that it has a Kasparov product and such that Lafforgues theory maps into it. With this theory, one can define a variant of the Bost conjecture with Banach algebraic coefficients and prove it in several important cases, e.g. if G has a gamma-Element that equals one in kk^{\ban}." |
| Alexander Gorokhovsky (University of Colorado - Boulder) |
March 27th, 2012 |
Title: Higher
analytic indices and symbolic index pairing Abstract: Higher index theory was started in the work of A. Connes and H. Moscovici on the Novikov conjecture. The goal of my talk is to reinterpret their theorem and to extend the higher index theory to new situations. This is joint work with H. Moscovici. |
| April 3rd, 2012 |
Canceled |
|
| Jianchao Wu (Vanderbilt University) |
April 10th, 2012 |
Title:
Equivariant Coarse Novikov Conjecture for Non-Positively
Curved Manifolds Abstract: When a locally compact group acts properly and isometrically on a Riemannian manifold, one can define an assembly map from the equivariant K-homology of the manifold to the K-theory of its equivariant Roe algebra. This map unifies the Baum-Connes assembly map and Roe's coarse assembly map. We will sketch a proof of the equivariant coarse Novikov conjecture, which concerns the injectivity of this map, in the case of non-positively curved and simply connected manifolds. |
| April 17th, 2012 |
Canceled |
|
| Remi Coulon (Vanderbilt University) |
April 24th, 2012 |
Title:
Random groups and Gromov's monster. Abstract: The idea of random groups has been introduced to answer the following question: how does a generic group look like? In this talk I will present some results about Gromov's density model of random group. Then I will explain how these ideas can be used to construct Gromov's monster group, a finitely generated group with surprising properties. |
Past events: link