# Operator algebra seminar

Speaker: Vito Felice Zenobi (Montpellier)

Title: Adiabatic groupoid and secondary invariants

Abstract: Let $G$ be a Lie groupoid.
I’ll define higher secondary invariant as classes in the K-theory of $C^*(G_{ad}^\circ)$,
the C*-algebra of the adiabatic deformation of $G$.
We’ll see how defining a wrong-way functoriality between these groups.
Moreover I’ll present the proof of the “Delocalized APS index Theorem”, related to cobordism formulas for these invariants.
Finally, if we have time, I’ll give product formulas between K-homology classes and secondary invariants.
All that applies to the study of  homotopy equivalences of Lie groupoids (through the signature operator)
and the study of metrics with positive scalar curvature along the $s$-fiber of the $G$.