Modular geometry on noncommutative tori
April 15-16, 2019
W303 School of Mathematics, Sichuan University
Yang Liu (MPI Bonn, Germany)
Noncommutative tori are probably the most well-known examples of noncommutative spaces. For instance, noncommutative 2-tori arise from actions on the circles by irrational rotations. Following the seminal work of Connes-Tretkoff and Connes-Moscovici there is a recent upsurge of activity on defining and computing notions of curvatures on noncommutative tori via the calculations of the coefficients of heat kernel asymptotics for the corresponding Laplace operators.
The computations are totally different from that for ordinary Riemannian manifolds. The noncommutativity prevents the analogue of the volume form to define a trace. This result in the appearance of a group of modular automorphisms, which is non-trivial even under conformal deformations of the flat geometry.
In this mini-course I will present the state of the art on “modular curvatures” on noncommutative tori. In particular, I will present relationships with hypergeometric functions. If time is allowed I will also present some relationships with cyclic theory.
Bohui Chen (Sichuan University)
Xiaojun Chen (Sichuan University)
Wenchuan Hu (Sichuan University)
Raphael Ponge (Sichuan University)
School of Mathematics, Sichuan University
Tianyuan Mathematical Center in Southwest China