SCIENTIFIC PROGRAMS AND ACTIVITIES

Upcoming talks 2009-10

TBA

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.