National and Kapodistrian University of Athens Noncommutative Geometry and Representation Theory March 6-10, 2023, Athens, Greece. Organised in Partnership with the Clay Mathematics Institute

Cohomology of polynomial growth, extension of cyclic cocycles and numeric invariants of Dirac operators

Not scheduled

50m

Speaker

Prof. Paolo Piazza (University Roma 1 Sapienza)

Description

In this talk I will present results showing the crucial role that geometric cycles of polynomial growth on groups play in the extension problem for cyclic cocycles on Galois coverings and on G-proper manifolds, with G a real reductive Lie group. The extendability property of these cyclic cocycles, from convolution algebras of functions of compact support to dense holomorphically closed subalgebras of suitable $C^{\ast }$-algebras, allows to define primary and secondary invariants of Dirac operators on these geometric structures. These invariants can in turn be used to solve purely geometric problems and I will survey some of them during the talk. This is joint work with Thomas Schick and Vito Zenobi for the part on Galois coverings and Hessel Posthuma for the part on G-proper manifolds. I will also talk briefly about delocalized invariants on G-proper manifolds leaving the detailed discussion to the talk of Hessel Posthuma. This last part is joint work with Posthuma, Yanli Song and Xiang Tang.

Presentation materials

Piazza Athens2023.pdf