CMA / AMSI Workshop

Noncommutative Geometry and Index Theory

22 - 27 July & 28 July - 1 August 2005

Workshop

Thursday 28 July - Monday 1 August 2005

Abstracts

Alan Carey
Semifinite local index theorem in noncommutative geometry

The talk is about some aspects of a new approach to the Connes-Moscovici local index theorem which enables an extension to spectral triples associated with a semifinite von Neumann algebra.

Public Lecture & MSI Colloquium
Joachim Cuntz
Index theorems of Atiyah-Singer type in the light of bivariant K-theory.

Joachim Cuntz
Stabilization by Schatten ideals and homotopy invariance

Piotr M. Hajac
Non-crossed-product examples of principal extensions of C*-algebras

The aim of this talk is to examine examples of C*-algebras equipped with a free action of U(1) that are not crossed products with their fixed-point subalgebras. We prove that a given C*-algebra is not a crossed product by determining appropriate K-invariant (Fredholm index) of a finitely generated projective module associated to the U(1)-action. This is in analogy with the classical fact that, if a principal bundle admits an associated vector bundle which is not trivial, then it itself cannot be trivial.

Lars Hesselholt
Bi-relative algebraic K-theory and topological cyclic homology

This is joint work with T. Geisser. It was recently proved by Cortiñas that, rationally, bi-relative algebraic K-theory and bi-relative cyclic homology agree. We show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application of this general theorem, we show that for a possibly singular curve over a field k of positive characteristic p, the p-adic algebraic K-groups and the p-adic topological cyclic homology groups agree in degrees greater than or equal to r where [k : kp] = pr. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology.

Nigel Higson
Index theory for SL(3)

The computation of C*-algebra K-theory for discrete subgroups of SL(3,C) is a leading issue in the area surrounding the Baum-Connes conjecture. As a first step toward approaching this problem, my student Bob Yuncken has been studying the construction of an index one operator on the flag variety for SL(3,C) which is analogous to the operator used by Kasparov in his treatment of SL(2,C). I shall give an account of this work, which blends geometry, representation theory and analysis.

Max Karoubi
Noncommutative methods in algebraic topology

Download abstract as PDF.

Tomasz Maszczyk
A pairing between super Lie-Rinehart and periodic cyclic homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology.

This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

Ryszard Nest
Geometry of Fourier integral operators with applications to index theorems

Adam Rennie
The Noncommutative Geometry of Graph C*-Algebras

Graph C*-algbras and k-graph C*-algebras provide an extremely large and diverse class of C*-algebras. They have been widely studied in part because they are amenable to computation. In this talk I will describe joint work with David Pask and Aidan Sims which shows that for a large class of these algebras, we can construct spectral triples and compute indices.

The novel feature of these examples is that they are naturally semifinite-meaning that we do not use the operator trace on B(H), but a different semifinite trace. The trace we use arises naturally from the geometry of the graph. The resulting semifinite index is an invariant of a finer structure than the isomorphism class of the algebra.

Jonathan Rosenberg
T-duality in string theory through ordinary topology and noncommutative geometry

An idea which is now well established in the physics literature is that "charges" on "branes" should take values in twisted (topological) K-theory, where the twisting is given by a cohomology class that represents the field strength. It is also expected that "T-duality" should hold, meaning that the theory on one space-time (with background field) is equivalent to that on another, where tori are replaced by their duals. I will describe recent joint work with Mathai Varghese in which we show how to make this rigorous for space-times which are principal torus bundles. Surprising conclusions are that sometimes the T-dual of a torus bundle turns out to involve non-commutative tori, and that often the set of torus bundles with background fields over a fixed base has a large automorphsism group (known to physicists as the "T-duality group").

Mathai Varghese
Twisted K-theory, twisted cohomology and cyclic homology

Bai-Ling Wang
D-brane, Fusion and Twisted K-theory

Twisted K-theory, in various form defined by Donavan-Karoubi in 1970 using bundles of simple algebras and Rosenberg in 1989 using continuous trace C*-algebras, have attracted many attentions partly as physicists predicted that it classifies the D-brane charges and represents new geometric features of the stringy space-time, but mainly mathematicians now know more about the underlying geometry of gerbes.

Given a D-brane, it is not known how to associate a twisted K-element, yet physicists apply ideas from conformal field theory to study some nice structures about twisted K-theory. A recent beautiful theorem of Freed-Hopkins-Teleman connects the equivariant twisted K-theory of a Lie group with the Verlinde ring of projective representations of the loop group. The underlying ring structure is still mysterious.

I will answer some of these issues using Thom isomorphism and Push-forward map in twisted K-theory, and Hamiltonian loop group actions. We can take away the word "should" from some of physicists' statements. The main tools are from classical differential geometry, with some ideas coming from C*-algebraic approachs.

Reference: Based on A. Carey & B. Wang's work on
1. Fusion of symmetric D-branes and Verlinde rings. math-ph/0505040.
2. Thom isomorphism and Push-forward map in twisted K-theory. math.KT/0507414
3. Multiplicative bundle gerbe and equivariant twisted K-theory. Work in progress.

Mariusz Wodzicki
Manifold aspects of the algebra of differential operators

My talk will serve a dual purpose: it will survey a number of past results and will offer several proposals for the future research.