Abstract

The classical Gauss-Bonnet theorem states that the total curvature of a closed Riemannian surface is a topological invariant. In this series of 3 lectures I shall first give an outline of an spectral formulation and proof of the Gauss-Bonnet theorem for the noncommutative two torus due to Connes and Tretkoff. Then I shall explain our ongoing work (joint with F. Fathizadeh) where this result is generalized to arbitrary values of the complex structure on the noncommutative two torus. This involves, in an essential way, some computer calculations with the heat kernel and noncommutative pseudodiffential calculus. The first talk will be a colloquium and shall serve as a general introduction to some of the pertinent ideas in this program.




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