Current research

My current research with my students and other co-workers focuses on Hilbert functions of admissible filtrations of ideals in Cohen-Macaulay rings. These rings are central objects of study in commutative algebra. Through the knowledge of their Hilbert polynomials we can obtain information about various blow-up algebras such as Rees algebras, associated graded rings, symmetric algebra and the ring itself. Information about these algebras is often useful in resolution of singularities as demonstrated in the works of Abhyankar, Hironaka, Rees, Sally, Zariski and others. 

We focus on their Cohen-Macaulay property. If a blow-up algebra is Cohen-Macaulay then computation of their algebraic and geometric invariants becomes easy since  several  homology and cohomology modules associated with them vanish. One of the techniques we use is the analysis of local cohomology modules of blow-up algebras. 

Local cohomology modules were introduced by A. Grothendieck in the fifties. These modules contain useful information which helps us in understating Hilbert polynomials as well as the Blow-up algebras. Specifically we use an avatar of the famous formula of Grothendieck and Serre which gives difference of Hilbert function and the Hilbert polynomial in terms of the Euler Characteristic of all the local cohomology modules of Rees algebra.

Our recent research using this  line of attack has yielded unified proofs of several classical and modern results about completeness of products of complete ideals in regular local rings proved by Zariski, Lipman-Teissier, Rees and Reid-Roberts-Vitulli, The work of Reid-Roberts-Vitulli used convex geometry and hence is applicable to only polynomial rings. We have been able to isolate a homological obstruction in terms of Castelnuovo-Mumford regularity which enables us to find a general result in all dimensions in analytically unratified local rings which yields a unified proof of several earlier theorems.