Giovanni Felder's lecture (6 hours, from Monday, Feb 5 to March 2, 2007)
Title: Hochschild homology of differential operators and
Riemann-Roch-Hirzebruch formula.
Abstract:
In 1989 Feigin and Tsygan proposed an algebraic approach
to theRiemann-Roch-Hirzebruch (RRH) formula, based on the
Hochschild homology of the algebra of polynomial differential
operators. In particular they constructed a distinguished class
in the cohomology of the Lie algebra of polynomial vector fields
mapping via formal differential geometry to the RRH
characteristic class. Recent progress in this subject is based on
new developments in deformation quantization, in particular the
Tsygan formality conjecture proved by Shoikhet. We will review
this theory and discuss some recent applications to traces in
deformation quantization and to extensions of the RRH theorem.