| Speaker |
Date |
Title and Abstract |
Jianchao Wu
(Vanderbilt University)
|
Sep 10, 2012 |
Title: Asymptotic Morphisms and the Atiyah-Singer index theorem
Abstract: This is the first in a series of introductory talks suitable for everyone with a basic knowledge of operator algebras and differential geometry. We will introduce asymptotic morphisms, a powerful tool in studying homomorphisms between K-theory groups. As an application, we will sketch a proof of the celebrated Atiyah-Singer index theorem, which is considered the starting point of noncommutative geometry. The follow-up talks in the future will discuss the Baum-Connes conjecture and its implications. |
Jianchao Wu
(Vanderbilt University)
|
Sep 17, 2012 |
Title: Asymptotic Morphisms and the Atiyah-Singer index theorem, continued
Abstract: We are going to continue our discussion of the Atiyah-Singer index theorem. We will demonstrate how ideas from noncommutative geometry can be applied to prove this classic result.
|
Xiang Tang
(Washington University)
|
Sep 21, 2012 (Friday)
2:10pm - 3:00pm
SC 1312
|
Title: Relative index and K-theory
Abstract: We will explain two interesting index Theorems (the Bojarski theorem and Atiyah-Weinstein-Epstein theorem) about the essential codimension of two projections with infinite dimensional range and kernel. We will discuss our attempt to understand these theorems using noncommutative geometry and K-theory tools. This is work in progress with Ronald Douglas and Jerome Kaminker. |
Gennadi Kasparov
(Vanderbilt University) |
Oct 1, 2012 |
Title: An introduction to the Baum-Connes conjecture
Abstract: The Baum-Connes conjecture is one of the most famous conjectures in noncommutative geometry. Roughly speaking, it provides a link between group homology and representation theory. This is especially interesting in the case of discrete groups because little is known about their representation theory. The talk will contain the statement, known results and applications of the Baum-Connes conjecture.
The talk will be accessible to graduate students with a basic knowledge of operator algebras. |
Stanley Chang
(Wellesley College)
|
Oct 8, 2012 |
Title: Curvature and rigidity: results of surgery
Abstract: In this talk, we will discuss the ways in which surgery theory is used in understanding the curvature and rigidity of compact and noncompact manifolds. We will explain the current state of knowledge about these ideas with respect to arithmetic manifolds.
|
Gennadi Kasparov
(Vanderbilt University) |
Oct 15, 2012 |
Title: An introduction to the Baum-Connes conjecture, continued
Abstract: This will be a continuation of the talk on October 1.
The Baum-Connes conjecture is one of the most famous conjectures in noncommutative geometry. Roughly speaking, it provides a link between group homology and representation theory. This is especially interesting in the case of discrete groups because little is known about their representation theory. The talk will contain the statement, known results and applications of the Baum-Connes conjecture.
The talk will be accessible to graduate students with a basic knowledge of operator algebras. |
Xin Li
(University of Münster, Germany)
|
Oct 22, 2012 |
Title: K-theory for semigroup C*-algebras
Abstract: The (reduced) semigroup C*-algebra of a left cancellative semigroup is simply given by the C*-algebra generated by the left regular representation of the semigroup. In this talk, I will explain how to compute K-theory for such
semigroup C*-algebras. It turns out that the Baum-Connes conjecture plays an important role in our computations. |
| |
Oct 29, 2012 |
No talk |
Fan Fei Chong
(Vanderbilt University) |
Nov 5, 2012 |
Title: On Atiyah's L^2-index theorem
Abstract: Let \Gamma be a discrete group acting properly and freely on a manifold M\tilde with compact quotient M = M\tilde / \Gamma. Given D any elliptic differential operator on the compact manifold M. Let D\tilde be the lifting of D to M\tilde. We may define the higher index of D\tilde, denoted by H-index(D\tilde), as an element of the K-group for the reduced C*-algebra of \Gamma K_0(C^*_r(\Gamma)). Let tr_* be the homomorphism on K-theory induced from the canonical trace on C^*_r(\Gamma). Then Atiyah's L^2-index theorem implies the following identity:
tr_*(H-index(D\tilde)) = index(D)
It follows that the Fredholm index of D is the 0-dimensional information of the higher index of D\tilde. I am going to sketch a proof of the Atiyah's L^2-index theorem via Geometric K-homology, as described in the joint paper by Willett and Yu in 2012. |
Jianchao Wu
(Vanderbilt University) |
Nov 9, 2012
(Friday)
2:10pm - 3:00pm
SC 1210
|
Title: The Novikov conjecture for groups of volume-form-preserving diffeomorphisms
Abstract: For a compact Riemannian manifold, we can construct an action of the group of its volume-form-preserving diffeomorphisms on an infinite-dimensional Hilbert manifold with non-positive sectional curvatures. Exploiting this geometric property, we can study the injectivity of the Baum-Connes assembly map for discrete subgroups of the volume-form-preserving diffeomorphism group. |
Jianchao Wu
(Vanderbilt University) |
Nov 12, 2012 |
Title: The Novikov conjecture and Hilbert manifolds with non-positive sectional curvatures
Abstract: We study the injectivity of the Baum-Connes assembly map for a group that acts isometrically and metrically properly on a simply connected Hilbert manifold with non-positive sectional curvatures. I will recapitulate last Firday's talk, before outlining a proof. Time permitting, I will present some interesting examples from geometry. |
