The core of an ideal in a commutative ring encodes information about all possible reductions of the ideal. It also has a close connection to Briancon-Skoda type theorems and to a conjecture by Kawamata about sections of line bundles. The proposer intends to further explore this interplay by studying the relation between cores and adjoints or multiplier ideals. Having worked on a formula for the core in equicharacteristic zero, he wishes to obtain a similar explicit expression in positive characteristic, where the shape of the core is markedly different. Likewise he would like to find a combinatorial description for the core of monomial ideals. The investigator plans to continue his work on blowup algebras of ideals, most notably of zero-dimensional ideals in regular local rings. He asks whether the quasi-Gorenstein property of the extended Rees algebra implies the Gorensteinness of the associated graded ring. He also suggests that the normality or Cohen-Macaulayness of the special fiber ring of a Gorenstein ideal may force the ideal to be a complete intersection. On a more computational note, he addresses the problem of constructing the integral closure of algebras, in particular of Rees algebras. Passing to the integral closure of the Rees algebra of an ideal is the first step towards resolution of singularities and the only known general method for computing the integral closure of the ideal. The proposer wishes to estimate the complexity of this process by finding bounds on the number of generators of the integral closure, the degrees of the generators and the number of steps required in the computation. As another measure of complexity he plans to study the Castelnuovo-Mumford regularity of powers and symmetric powers of homogeneous ideals having dimension at most one. He expects that estimates on the regularity do not only persist when the ideal is raised to powers, but that they actually improve. Similar improved bounds for the regularity of symmetric powers would help finding the equations of Rees algebras and thereby lead to efficient algorithms in elimination theory. The proposer also intends to continue his work on generalized principal ideal theorems. The goal is to bound the codimension and prove connectedness properties for degeneracy loci of maps of modules that are not necessarily free; here one has to assume that the maps are not `too generic'. The investigator proposes a weak version of this condition by introducing a notion of ampleness for modules over local rings. He hopes to prove principal ideal theorems that only require the weaker assumption, thus generalizing the known results in both local algebra and projective geometry.
The investigator works in Commutative Algebra, an area concerned with the qualitative study of systems of polynomial equations in several variables. Such systems arise in numerous applications outside of mathematics. Over the past two decades commutative algebraists have become increasingly interested in computational aspects, thereby emphasizing connections to applied areas such as computer algebra, robotics, cryptography and coding theory. This investigator's research too has a strong computational component. Part of the project involving the collaboration with mathematicians in Brazil is funded by the NSF Office of International Science and Engineering
