My research work is primarily in Algebraic Geometry which studies the geometric properties of solutions of systems of polynomial equations. In some cases, this also leads to problems in Differential Geometry, Algebraic Topology and Number Theory; hence I have some papers in these areas as well.
The contributions that would count as significant are:
1. A restatement (with S. Ramanan) of Green's conjecture (which relates the syzygies of a curve embedded via its canonical linear system (an algebraic invariant) with the existence of linearly dependent points in this curve (a geometric invariant)) in terms of a question about vector bundles.
2. The proof of the Hodge Conjecture (a millenium prize problem) in some special cases. Specifically, we produced the first example of the Kuga-Satake correspondence for a lattice of rank 6. Secondly, we proved (with I. Biswas) the Hodge Conjecture for generic member of various families of Abelian varieties which are not defined by group theoretic considerations.
3. A statement of certain explicit cycle-theoretic conjectures for hypersurfaces and complete intersections. These conjectures are consequences of the much broader conjectures of Bloch and Beilinson. We also proved these conjectures when the dimension is very large as compared with the degree.
4. Following an influential paper by Lazarsfeld we explored (with V. Srinivas) self-maps of Grassmannians and maps from Quadrics to other spaces.
