Several central open questions concerning tight closure in positive characteristic, such as whether tight closure commutes with localization, and whether the tight closure of an ideal of a domain coincides, in good cases, with the contracted expansion of the ideal from the absolute integral closure of the domain are proposed for study. Another main thrust is to explore several notions of closure in rings that do not contain a field, such as finitely generated algebras over the integers, with the hope of extending tight closure theory to such rings, thereby solving many open questions. A new approach to the problem of proving existence of big Cohen-Macaulay modules in mixed characteristic will be pursued, as well as several lines of research aimed at solving the long standing and important question of whether regular rings are direct summands of their module-finite extensions.
Commutative rings are abstract systems in which one can perform addition, subtraction and multiplication. The integers and real or complex numbers are examples, but there are vastly different sorts of rings as well, including finite rings. One can introduce variable elements into any ring, forming a larger ring. Ring theory can be used to study the behavior of large systems of equations in many variables. There are methods of transition from the study of equations with real or complex coefficients to the study of related systems with coefficients in a finite ring. These methods produce qualtitative information about the solutions of the original systems of equations: sometimes one can determine whether there exist solutions and, if so, what the dimension of the solution space is. Most of the problems in the proposal can be viewed as problems about solving equations. The projects in the proposal will provide fundamental information about the nature of the solutions for many sorts of systems of equations.
