Contact information:

Phone: (517) 353-8145
Fax: (517) 432-1562
Email: kulkarni at math dot msu dot edu
Office: D303 Wells Hall

Research

The research described below (except for the first two articles) has been partially supported by the NSF (DMS 0202295, 0311850 and 0603684). The research in last two articles has also been suppoorted by an IRGP grant from the Michigan State University.

My research involves interactions between algebra, algebraic geometry and topology. Here is a brief description of various projects I have worked on.

Noncommutative algebra and algebraic geometry

All of my projects in this area study various families of algebras. A distinction can be made between these families based on whether or not the algebras in these families are finitely generated modules over their centers.

Deformations of Lie algebras

In two projects in this area, I studied two families of algebras. Thefirst was a 7-parameter deformation of U(sl_2). The origins of this family were in physics and they were introduced by E. Witten. The goals of this project were to classify finite dimensionalirreducible modules of these algebras and to classify infinite dimensional irreducible modules for generic values of parameters.

The second family had combinatorial origins and was introduced by G. Benkart and T. Roby. For these algebras, I computed their center and classified their finite (and in most cases infinite dimensional) irreducible representations. For roots ofunity like cases, I studied these algebras in more detail to obtain further structural results.

Algebras which are finite modules over their centers

There are two projects in this section that I have worked on.The first project studies Clifford algebras ofbinary forms. As in the case of quadratic forms, one can construct algebras which are universal for linearizations of forms of higher degree. These algebras are important for the purposes of studying higher degree forms and have interesting, fruitful connections with the arithmetic of the underlying field. In my thesis, I showed that the moduli space of dimension d representations of th eClifford algebra of a binary form of degree d is the complement of a theta divisor in the Picard variety (of some degree) of a smooth projective curve canonically associated to the form. This was done for forms over arbitrary fields with mild conditions. The hope was that this may then have some arithmetic consequences. In a subsequent article, I have made progress in this direction. In this article, I give a condition(on some principal homogeneous space to be trivial) for the reduced Clifford algebra to be split. (The reduced Clifford algebra is a quotient of the Clifford algebra, is Azumaya over its center and these two algebras have the same d dimensional representations.)

In joint projects with Daniel Chan , we consider sheaves of maximal orders on projective surfaces. Over curves, or locally over discrete valuation rings, there is a beautiful arithmetic theory of maximal orders. (For example, see Reiner's book.) Over surfaces this theory is more involved. It was initiated by several papers of M. Artin in the '80s (in the local case). In this approach, we reduce the questions of maximal orders to the geometry of the underlying surface. The idea (from an etale cohomology sequence of M. Artin and D. Mumford) is that one can construct a ramification divisor which a collection of irreducible curves with their (possibly branched) cyclic covers. This then reflects the order geometrically, though it does not determine it. In the first article, we consider orders called del Pezzo orders on surfaces and give classification of theirramification data. In the second article, we classified orders called numerically Calabi-Yau orders on surfaces. One can define Kodaira dimension of orders using the canonical sheaf on the order. Then currently we are working on orders of Kodaira dimension 1. We are also studying line bundles on a del Pezzo order.

Intersection homology and representation theory

I have also been interested in representation theoretic questions involving algebraic groups and symmetric spaces. Some spaces which occur naturally in these questions are not manifolds, but their singularities are not too bad. Poincare duality fails tohold in general for these stratified spaces. In the 70's, M. Goresky and R. MacPherson developed the theory of intersection homology to account for this failure. Intersection homology (for lower/upper middle perversity)satisfies Poincare duality in many cases, notably for complex algebraic varieties. However this duality doesn't hold in general. Good examples of spaces for which intersection homology doesn't satisfy duality are reductive Borel-Serre compactifications of certain Hermitian symmetric spaces. In joint work with Markus Banagl, we show that a self-dual homology theory which iscompatible with intersection homology does exist for reductive Borel-Serre compactifications of Hilbert modular surfaces. We have further proved a similar result for Hilbert Blumenthal varieties and also have constructed examples (of Hermitian symmetric examples) where such a self-dual sheaf doesn't exist.

Here are more technical abstracts of my papers mentioned above: