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.

I will try to discuss how to extend the classical moment maps to the diffeological case, and hopefully some applications to some orbifolds, etc.

TBA

Abstract: Feynman's theorem, a fundamental mathematical tool in quantum field theory, provides a way to evaluate complicated integrals by summing over finite graphs. After exploring how to obtain this sum from an integral, we will reverse the correspondence and prove a famous result of Cayley, that the number of labelled trees with n vertices is nn−2, using known values of integrals. No physics will be needed to follow this lecture. For some background material please check out the Pizza Seminar blog http://pizzaseminaruwo.blogspot.ca/

I will discuss about my recent joint work with Yu. Berest and F. Eshmatov on the structure of automorphism groups of class of algebras closely related to the first Weyl algebra $A_1(k)$. In particular, we give a geometric presentation for these groups using the Bass-Serre theory, thus answering question posed by T. Stafford in 80's. A key role in our approach is played by a transitive action of the automorphism group $Aut(A_1)$ on the Calogero-Moser varieties, which are certain variation of Hilbert schemes of points on the plane. In particular, using this geometric description of these groups we have been able to classify these groups up to isomorphism. The results obtained generalize some classic theorems of J. Dixmier and L. Makar-Limanov. In the end, we discuss a few open problems related to these groups such as the Dixmier Conjecture.

Given a mathematical object, one may associate a combinatorial object that captures some of its properties.  Natural examples would be matroids as combinatorial abstractions of linear subspaces and Newton polytopes as combinatorial abstractions of hypersurfaces.  Then one has a class of combinatorial objects that behave somewhat like the original mathematical objects.  Two questions arise: which properties of the original objects do the combinatorial objects encode?; and how to characterize the combinatorial objects that are abstractions of an object in the original category. 

In this talk, we discuss tropical varieties which are combinatorial abstractions of algebraic varieties and contain as examples the theories of Newton polytopes and matroids.  We develop some examples and share the progress that has been made on those two questions.

The theory of $\it{enriched}$ categories generalizes naturally the classical theory of categories and has at least as much impact in mathematics as classical category theory. Following ideas of B$\mathrm{\acute{e}}$nabou, Grothendieck, Kelly, Mac Lane, Segal and others, we develop a theory of $\it{weakly}$ $\it{enriched}$ categories; these structures arise naturally when Algebra meets Homotopy and give an alternative approach to $\it{higher}$ $\it{categories}$. In this talk I will give an overview of classical enriched category theory and then will talk about the extension to $\it{weak}$ $\it{enrichment}$.