In this lecture we first give some examples of noncommutative C*-algebras. Then we introduce group C*-algebra of a discrete group. Finally we present one of the most important theorems in the theory of C*-algebras that is called GNS construction, by which we can see every C*-algebra as a norm closed subalgebra of the algebra of bounded linear operators on a Hilbert space.

I will discuss an extension of Chow groups to derived schemes and discuss how to recover Kontsevich's formulas relating virtual structure sheafs and virtual fundamental classes as a byproduct of the use of quasi-smooth morphisms between derived schemes.