Studies in Commutative Algebra and Computational Algebra

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This proposal concerns problems in commutative algebra, algebraic geometry and computational algebra. These problems range from the explicit construction of algorithms in algebra and the development of settings to analyze their complexity to the construction and analysis of several classes of singular varieties. The investigator is: (a) developing methods to construct integral closures of algebras, ideals and modules, and understanding their complexity, including the introduction of non-Turing models of complexity; (b) developing families of numerical signatures (multiplicities, volumes) of rings and algebras that play in local rings and arbitrary algebras a role similar to Castelnuovo--Mumford's regularity in graded structures; (c) carrying out an algebraization of blowup algebras and of algebras associated to commuting sets of elements of Lie algebras with the major aim of finding its properties of Cohen-Macaulay type (including rational singularity), arguably the most efficient packaging of an algebraic structure.The mathematical problems with which this proposal is concerned come from the overlapping areas of commutative algebra, algebraic geometry and computational algebra. The research program is focused on the search for generic and numerical solutions of sets of polynomial and analytic equations, such as those that apply to such diverse areas as algebraic geometry, combinatorics, cryptography, control/coding theory and robotic motion. The investigator studies the fine structure of these algebraic systems, and he develops methods and algorithms for solving them. At the same time he is seeking to break the computational logjam of several problems of computer algebra through a more fundamental understanding of their structure.