Abstract: To each differential-graded algebra and element a\in A^1, we associate a cochain complex, where the map is defined by the multiplication by a. The degree l resonance variety is the set of elements a in A^1 such that the l-th cohomology is not zero. It is shown that The degree l resonance variety, up to ambient linear isomorphism, is an invariant of A. The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The main motivation for the study of resonance varieties comes from the tangent cone, which there is a close relation between the degree-one resonance varieties to the characteristic varieties, where the tangent cone of W at 1 is the algebraic subset TC_1(W) of C^n defined by the initial ideal in(J) \subset S. In this talk we describe the degree-one resonance variety. We will be particularly interested in the resonance varieties of graphical arrangements.

Abstract: I will explain the construction of Kontsevich-Vishik canonical trace on non-integer order classical pseudodifferential operators. This construction has it roots in the old methods of extracting a finite part from a divergent sum or integral (infra-red and ultra-violet divergence), used by mathematicians and physicists. If time permits I will explain some of the results on generalizations of this construction to noncommutative setting.