Abstract: I will survey on a recent joint work with M. Khalkhali and a paper by Connes and Moscovici on scalar curvature for noncommutative two-tori. The scalar curvature is computed by considering small time heat kernel expansions of the perturbed Laplacian which encodes the metric information of a general translation invariant conformal structure and a Weyl conformal factor on the noncommutative two-torus. There is an equivalent formulation for the scalar curvature in terms of special values of spectral zeta functions. I will also talk about our result on the Gauss-Bonnet theorem for noncommutative two-tori which extends the work of Connes and Tretkoff to the general conformal structures.

Abstract: In the talk there will be given a short review of holomorphic extension problems starting with the famous Hartogs theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent results on global holomorphic extensions for unbounded domains obtained together with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago). The classical Hartogs theorem solves the extension problem for bounded domains in C^n and clearly shows the difference between one and many-variables cases. The theorem is considered as an informal beginning of Complex Analysis in Several Variables. Surprisingly, the unbounded case was missed by analysts for more than a hundred years, even though it is important not only in Complex Analysis, but also in Partial Differential Equations and Algebraic Geometry. The problem appeared highly non-trivial and the work is in progress. However the talk will be illustrated by many figures and pictures and should be accessible also to graduate students.