Ravi S. Kulkarni

Research Summary:

1. In the past few years I have been working on a general theory of dynamical types of transformations in classical geometries. This theory extends a part of Weyl's work on compact Lie groups to arbitrary actions. I reported the initial developments in earlier communications. Last year, I have worked out further details. A thesis by Rony Gouraige in City University of New York ( who visited HRI in 2002-03 ) entitled "Conjugacy Classes of Centralizers in the Algebra of Endomorphisms of a finite-dimensional vector space over a central division algebra", is a good illustration of this theory. The whole work will appear in the form of some papers and a monograph.


2. Frenet's curvatures for curves with variable speed, and arithmetic aspects of curvatures functions in pseudo-Riemannian geometry ( in preparation ).


3. ( Generalizing an old work of W. Blaschke, we study intrinsic and extrinsic curvature functions in pseudo-Riemannian geometry. A theme is : for pseudo-Riemannian manifolds admitting polynomial parametric equations with rational coefficients, the squares of curvature functions at rational points are rational numbers. )


4. ( with Satya Deo ) : Branched Coverings, Labelled Coverings, and Transitive permutation Groups. ( in preparation ).

Invited Lectures/Seminars :

1. University of Toronto, Toronto, Canada, November 2003.
2. Rutgers University, New Brunswick, NJ, November 2003.
3. Prof. C. S. Venkataraman Memorial Lecture, organized by the Ramanujan Mathematical Society, in Dr. B. R. Ambedkar University, Agra, July 2004.
4. Morningside Center of Mathematics, Chinese Academy of Sciences, August 2004.

Other Activities:

I participated in

1. the NBHM-sponsored Advanced Foundational School on Differential Topology, and Linear Algebra, at IIT Bombay, Powai, in May 2004 ( 4 lectures each ), and
2. Visiting Students Summer Programme, HRI, in June 2004 ( 8 lectures ).