Studies on Cores of Ideals and Blowup Algebras

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This project is concerned with problems from commutativealgebra and its interactions with computational algebra andalgebraic geometry. The focus of the proposal is on the theory ofblowup algebras. These algebras appear naturally in manyconstructions both in algebra and geometry. For example they arisein the process of resolution of singularities. An essential tool intheir study is the concept of reduction of an ideal. Several keyinvariants emerge in this context such as the reduction number,which among other things controls the Cohen-Macaulay property ofblowup algebras. To study all reductions at once one considers thecore of an ideal, defined as the intersection of these reductions.This object, related to coefficient, adjoint and multiplier ideals,plays a crucial role in Brian\c{c}on-Skoda type theorems. A mainthrust of this proposal is to give a combinatorial description ofthe core of monomials ideals, to clarify the connection of the corewith the reduction number, the first coefficient ideal and theadjoint or multiplier ideal, to investigate the core in arbitrarycharacteristic and to establish formulas that were previously knownonly in characteristic zero. In addition, the project also studiesother areas such as integral closures of ideals. The concepts ofintegral extensions and integral closures of rings and ideals arecentral to much of commutative algebra. One of the goals of theproject is to find a priori measures for the complexity of computingintegral closures.Often real life problems involve many unknown parameters that arerelated by equations which are impossible to solve exactly.Nevertheless, using commutative algebra methods, much valuabledescriptive information can be gained about the potential sets ofsolutions, if not the exact solutions themselves. Commutativealgebra has seen a great deal of activity and success over the pasttwo decades with the solution of important conjectures, itsapplication to diverse fields such as computer science,cryptography, coding theory, robotics, pattern recognition andtheoretical physics, and the discovery of unexpected connections toother parts of mathematics, ranging from topology to combinatoricsand from computer algebra to statistics.