Weakly parametric pseudodifferential calculus on Heisenberg modules

14.01.2022, 11:00  –  2.22 und online
Arbeitsgruppenseminar Analysis

Gihyun Lee (MPI, Bonn)

A decade ago, Connes-Tretkoff interpreted the concept of intrinsic curvature in terms of noncommutative geometric language. Connes-Tretkoff worked on noncommutative tori, which are important examples of noncommutative spaces in the sense of Alain Connes’ noncommutative geometry. Afterwards Lesch-Moscovici extended the results of Connes-Tretkoff to Heisenberg modules, the objects which were introduced to implement Morita equivalence relation between noncommutative tori. Having a pseudodifferential calculus on Heisenberg modules is of crucial importance in order to compute desired geometric information. For that reason, Lesch-Moscovici introduced a novel pseudodifferential calculus on Heisenberg modules, and applied it to compute and study the noncommutative analogue of Gaussian curvature.

Meanwhile, on a compact manifold, the noncommutative residue trace introduced by Wodzicki and Guillemin is a trace on the algebra of integer order pseudodifferential operators, and it plays an important role in geometry and mathematical physics. The essential part in the construction of the noncommutative residue trace is the derivation of the asymptotic expansion of the resolvent trace of an elliptic operator, and this asymptotic expansion can be derived by using Grubb-Seeley’s weakly parametric pseudodifferential calculus.

In order to pave the way to construct the analogue of the noncommutative residue trace on the algebra of pseudodifferential operators on Heisenberg modules along the same lines in the case of compact manifolds, it would be desirable to construct an analogue of weakly parametric pseudodifferential calculus and derive the asymptotic expansion of the resolvent trace of elliptic operators on Heisenberg modules. In this talk, I will give an overview of these constructions.

Based on joint work with M. Lesch.

For those of you who can and would like to join us, please meet us in the seminar Room 2.22 of the maths institute, where we can follow the talk together. Please let us know in advance if you plan to come to the institute.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks, please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.