Abstract: Studying the Ricci flow of a smooth, closed manifold M equipped with a Riemannian metric g involves the process of allowing the metric g to evolve over time under the PDE g_{t} = -2Ric(g). Ricci flow was, in fact, the main tool used by Perelman to prove the Poincare conjecture. The purpose of this talk will be to discuss what is Ricci flow, to present where it comes from and to provide examples of Ricci flow of certain manifolds. Our discussion will then lead into an analysis of a paper written by Bhuyain and Marcolli, who constructed a version of Ricci flow for noncommutative two-tori. The Ricci flow is a fundamental tool used to understand the geometry and topology of manifolds, and understanding it well will help us understand how we can classify other noncommutative spaces such as noncommutative tori in higher dimensions.

Abstract: Recently an analogue of the Gauss-Bonnet theorem has been proved by Connes-Tretkoff and Fathizadeh-Khalkhali for noncommutative two torus. The idea is based on the direct computation of the value at origin of the zeta function associated to the corresponding Laplacian. In this talk we will briefly discuss the above theorem and explain a related problem.