This seminar, organized by Man-Duen Choi, Chandler Davis, George Elliott, and Peter Rosenthal, is devoted, loosely speaking, to the study of all areas of mathematics involving Hilbert space operators. (This has variously been interpreted recently to include matrix theory, index theory of differential operators and other aspects of non-commutative geometry, invariant subspace theory, quantum information theory, C*-algebra classification theory, and string theory.)

In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer still holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulate the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the talk will be a review of noncommutative geometry and Connes and Moscovici's index theorem in diffeomorphism-invariant geometry. In the 2nd part, I will hint at on-going projects on the reformulation of the local index formula in two new geometric settings: the biholomorphism-invariant geometry of strictly pseudo-convex domains and the contactomorphism-invariant geometry of contact manifolds.

- · Monday, Jan. 25, 2010: K-theory of C*-algebras and KK-theory (Abhijnan Rej)
- · Monday, Feb. 01, 2010: Introduction to KK-theory (and its categorical properties) (Abhijnan Rej)
- · Monday, Feb. 08, 2010: The intersection product in KK-theory and non-commutative topology (Abhijnan Rej)
- · Monday, Feb. 22, 2010: D-branes and KK groups (Abhijnan Rej)
- · Monday, Mar. 22, 2010: Relative determinants and isospectrality on surfaces with cusps (Clara Aldana)
- · Monday, Mar. 29, 2010: The Cuntz semigroup of C(X,A) where A is a simple AF C*-algebra (Aaron Tikuisis)
- · Monday, Apr. 05, 2010: Basis map operators arising from regular semigroup representations (Barry Rowe)
- · Monday, Apr. 19, 2010: On the classification of C*-algebras (Henning Petzka)
- · Monday, May. 10, 2010: The Dirac operator on the noncommutative space of connections (Alan Lai)
- · Tuesday, May. 11, 2010: Naimark extensions for indeterminacy in the moment problem (Franek Szafraniec)
- · Monday, May. 31, 2010: On triangular factorization of positive operators (Lev A. Sakhnovich)
- · Monday, Sep. 20, 2010: Quantum error correction, and higher rank numerical ranges of normal matrices (Chi-Kwong Li)
- · Friday, Jan. 14, 2011: Spectral zeta functions and Gauss-Bonnet type theorems in noncommutative geometry (Masoud Khalkhali)
- · Monday, Feb. 07, 2011: Group actions and noncommutative geometry (Raphael Ponge )
- · Monday, Mar. 07, 2011: On certain multiplier projections (Henning Petzka)
- · Monday, Mar. 14, 2011: What are quantum channels through my old dream? (Man-Duen Choi)
- · Monday, Mar. 21, 2011: The nuclear dimension of a C* algebra (Nicola Watson)
- · Monday, Mar. 28, 2011: A generalization of the Weyl-von Neumann Theorem (Daniel Hay)
- · Monday, Mar. 19, 2012: Characterizing $\mathcal{Z}$-Stable $C^*$-algebras (Danny Hay)
- · Monday, Mar. 26, 2012: A stably finite $C^*$-algebra without bounded trace (Henning Petzka)
- · Monday, Apr. 02, 2012: Decomposition rank, nuclear dimension and tracial approximation (Nicola Watson)
- · Monday, Apr. 23, 2012: Hilbert Modules, Strong Morita Equivalence and Rotation Algebras (James Lutley)
- · Monday, May. 28, 2012: Random Projections, Free Probability, and Holomorphic PDEs (Todd Kemp)
- · Tuesday, Nov. 20, 2012: Universal C*-algebras of *-semigroups and the C*-algebra of a partial isometry (Berndt Brenken)
- · Monday, Dec. 17, 2012: Constructing subfactors with jellyfish (Dave Penneys)