Modular geometry on noncommutative tori

[TMCSC]

April 15-16, 2019

W303  School of Mathematics, Sichuan University

[minicourse I]0415-16Yang Liu-01.png

SPEAKER

Yang Liu (MPI Bonn, Germany)

abstract

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.

ORGANIZERS

Bohui Chen (Sichuan University)

Xiaojun Chen (Sichuan University)

Wenchuan Hu (Sichuan University)

Raphael Ponge (Sichuan University)

SUPPORTED BY

School of Mathematics, Sichuan University

Tianyuan Mathematical Center in Southwest China