There is a result from the 1980's that allows us to describe the equivariant $K$-theory of curves: if $X$ and $Y$ are curves, G is a finite reducible group and $Y = X/G$, then we can write $K_G(X)$ in terms of $K(Y)$ and some representation rings associated to the group. Prof. Dhillon and I have generalized this result to any dimension using the description of the category of coherent sheaves on a root stack given by Borne and Vistolli.

I will report on results obtained in a recent paper with Baran Serajelahi. I will describe quantization constructions, obtained from Berezin-Toeplitz quantization, for an n-dimensional compact Kahler manifold regarded as a (2n-1)-plectic manifold, and for a hyperkahler manifold.

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.

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.

In this talk, we will explore few logical foundations for quantum theory. I will briefly introduce logical syntax and semantic, and the important notion of Lindebaum Algebra. After acquiring essential logical equipment, I will discuss Partial Boolean Algebras as an algebraic model to formulate Kochen- Specker theorem in pure logical setting. I will as well mention the toposical foundation for quantum theory and the formulation of Kochen- Specker theorem in this setting. During the talk, some historical and philosophical issues will also be addressed.

We will present a theorem due to J. Cohen that the stable homotopy groups of spheres are generated under higher Toda brackets by the classes in Adams filtration one: the Hopf classes as well as the first alpha element (for odd primes).

We construct a combinatorial framework for proving cohomological vanishing results on certain classes of spaces, by means of a Mayer-Vietoris-type spectral sequence and certain Cohen-Macaulayness hypotheses. The spaces include complex hyperplane complements, their De Concini-Procesi compactifications, and configuration spaces of points in tori. In particular, we generalize classical vanishing results due to Kohno, Esnault, Schechtman and Vieweg, and recent work of Davis, Januszkiewicz, Leary and Okun.

This is joint work with Alex Suciu and Sergey Yuzvinsky.

After a quick review of the general theory of the Dirac operators and fixing some notations, I will construct the Dirac operator for some of Riemannian manifolds. This will include torus with conformally perturbed flat metric, 2-sphere and 3-spheres with the round metric and if time allows me, I will construct the Dirac operator for the Robertson-Walker metrics.