April 14, 2024

Bimonthly Canadian Noncommutative Geometry Workshop 2009-10
at the Fields Institute.

This bimonthly workshop aims to cover new developments in Noncommutative Geometry, and each workshop features a keynote address by one of the top people in the field. This workshop is associated with the Center for Noncommutative Geometry and Topology, University of New Brunswick and the Department of Mathematics at the University of Western Ontario.
Organizers: Masoud Khalkhali, Dan Kucerovsky, Bahram Rangipour

Upcoming talks 2009-10



Past Workshops

Oct. 3, 2009
10:30 am
Sheldon Joyner (University of Western Ontario)
The geometry of the functional equation of Riemann's zeta function

In a seminal 1859 paper, Riemann gave two proofs of the analytic continuation and functional equation of his zeta function. The ideas behind his theta function proof were later developed into a powerful theory of Fourier analysis on number fields, in work of Hecke, Tate and others. In this talk, I will focus instead on the contour integral proof, and based on the ideas therein, will present two infinite families of new proofs of the analytic continuation and functional equation. The proofs are facilitated by geometric data coming from the fact that the polylogarithm generating function is a flat section of the universal unipotent bundle with connection over P^1\{0, 1, \infty}.

1:30 pm
Frederic Rochon (University of Toronto)
Eta forms and the odd pseudodifferential index

In this talk, we will introduce eta forms for a general family of elliptic self-adjoint pseudodifferential operators. These eta forms can intuitively be understood as a regularized version of the Chern character. Although their definition is not local, we will see that their exterior differential is and in fact represents the Chern character of the odd index of the family. The proof will be topological in nature and will involve a short exact sequence of groups of pseudodifferential operators that are classifying spaces for K-theory. If time permits, we will also explain how these eta forms can be used to compute the curving of a certain bundle gerbe naturally associated to the family. This is a joint work with Richard Melrose.


Aug. 15, 2009
11:00 am

Raphael Ponge (University of Tokyo)
On the Noncommutative Residue and Dixmier Trace
The Dixmier trace plays a central role in noncommutative geometry since it is the noncommutative subsitute for the classical integral. The noncommutative residue is a trace on pseudodifferential operators which was found independently by Guillemin and Wodzicki in the early 80s. It appears as the residual trace induced on pseudodifferential operators of integer orders by the analytic extension of the usual trace to pseudodifferential operators of non-integer orders.
A well-known result of Connes says that on a compact manifold of dimension n, the noncommutative residue agrees with the Dixmier trace on pseudodifferential operators of order less than or equal to -n. Thus the noncommutative residue allows us to extend the Dixmier trace to all pseudodifferential operators, even to those that are unbounded.
In the first part of the talk, after a review of the constructions of the noncommutative residue and Dixmier trace, a new and elementary proof of Connes' theorem will be given. This proof is essentially a reduction to Rn and avoid any use of complex powers of an elliptic operators. In particular, unlike other proofs, it holds in wide variety of other settings.
In the second part of the talk, we will discuss the extension of this proof to the Heisenberg calculus and how this enables us to compute the Dixmier trace of commutators of Toeplitz operators on contact manifolds.
If time permitted, we will also discuss the construction of the contact and CR invariants using noncommutative residues of geometric projections, which is mostly motivated by Fefferman's program.

Aug. 15, 2009
2:00 pm
Piotr Hajac (University of Warsaw)
Toeplitz Quantum Projective Spaces
We define the C*-algebra of a quantum complex projective space TP(n) as a multirestricted fiber product build from (n+1)-copies of the n-th tensor power of the Toeplitz algebra (Toeplitz cubes). Replacing the Toeplitz algebra by the algebra of continuous functions on a disc, one obtains the algebra of continuous functions on CP(n). Using Birkhoff's theorem on distributive lattices, we show that the lattice generated by the ideals defining this fibre product is free. This means that the fiber product structure is "maximally non-trivial" or, in geometric terms, that all possible intersections obtained from pieces of Toeplitz cubes covering this quantum projective space are non-empty. This is a property inherited from the affine covering of a projective space. All this is used as an example to illustrate the classification of finite closed coverings of compact quantum spaces by finitely supported flabby sheaves of algebras over the universal partition space (the infinite projective space over Z/2 equipped with the Alexandrov topology). Based on joint work with Atabey Kaygun and Bartosz Zielinski.

Support for graduate students is available, please enquire, ncgworkshop<at>
This workshop is associated with the Center for Noncommutative Geometry and Topology at the University of New Brunswick and the Noncommutative Geometry Group at the University of Western Ontario.
We thank the Fields Instutute for financial support.

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