This project is concerned with problems from commutative algebra and its interactions with computational algebra and algebraic geometry. The focus of the proposal is on the theory of blowup algebras. These algebras appear naturally in many constructions both in algebra and geometry. For example they arise in the process of resolution of singularities. An essential tool in their study is the concept of reduction of an ideal. Several key invariants emerge in this context such as the reduction number, which among other things controls the Cohen-Macaulay property of blowup algebras. To study all reductions at once one considers the core of an ideal, defined as the intersection of these reductions. This object, related to coefficient, adjoint and multiplier ideals, plays a crucial role in Brianc{c}on-Skoda type theorems. A main thrust of this proposal is to give a combinatorial description of the core of monomials ideals, to clarify the connection of the core with the reduction number, the first coefficient ideal and the adjoint or multiplier ideal, to investigate the core in arbitrary characteristic and to establish formulas that were previously known only in characteristic zero. In addition, the project also studies other areas such as integral closures of ideals. The concepts of integral extensions and integral closures of rings and ideals are central to much of commutative algebra. One of the goals of the project is to find a priori measures for the complexity of computing integral closures.
Often real life problems involve many unknown parameters that are related by equations which are impossible to solve exactly. Nevertheless, using commutative algebra methods, much valuable descriptive information can be gained about the potential sets of solutions, if not the exact solutions themselves. Commutative algebra has seen a great deal of activity and success over the past two decades with the solution of important conjectures, its application to diverse fields such as computer science, cryptography, coding theory, robotics, pattern recognition and theoretical physics, and the discovery of unexpected connections to other parts of mathematics, ranging from topology to combinatorics and from computer algebra to statistics.
