Studies in Commutative Algebra and Computational Algebra

这个建议涉及交换代数，代数几何和计算代数中的问题。 这些问题的范围从明确建设的算法在代数和发展的设置，以分析其复杂性的建设和分析几类奇异品种。 研究者是：（a）开发构造代数、理想和模的整数闭包的方法，并理解它们的复杂性，包括引入复杂性的非图灵模型;（B）开发数字签名族（多重性，体积）的环和代数，发挥在当地的环和任意代数的作用类似于Castelnuovo-芒福德的规则性分级结构;（c）对爆破代数和与李代数的元素交换集相关的代数进行代数化，主要目的是找到其Cohen-Macaulay型性质（包括理性奇点），可以说是代数结构的最有效的包装。这个建议所涉及的数学问题来自重叠的领域交换代数，代数几何和计算代数。 该研究计划的重点是寻找多项式和解析方程组的通用和数值解，例如适用于代数几何，组合数学，密码学，控制/编码理论和机器人运动等不同领域的方程组。 研究者研究这些代数系统的精细结构，并开发解决这些问题的方法和算法。 与此同时，他正在寻求打破计算僵局的几个问题的计算机代数通过更根本的理解他们的结构。

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.