Studies in Commutative Algebra and Computational Algebra

这一建议涉及交换代数、代数几何和计算代数中的问题。这些问题的范围从代数算法的显式构造和分析其复杂性的设置的发展到几类奇异簇的构造和分析。研究人员是：(A)发展构造代数、理想和模的积分闭包的方法，并了解它们的复杂性，包括引入复杂性的非图灵模型；(B)发展在局部环和任意代数中发挥类似于Castelnuovo-Mumford在分次结构中的正则性的作用的环和代数族的数字签名(重数、体积)；(C)对爆破代数和与李代数的交换元素集相关的代数进行代数化，主要目的是找出它的Cohen-Macaulay型(包括有理奇点)的性质，这是代数结构的最有效的包装。该研究计划专注于寻找多项式和解析方程组的通用和数值解，例如那些适用于代数几何、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。研究人员研究了这些代数系统的精细结构，并开发了求解它们的方法和算法。与此同时，他正在寻求通过对计算机代数结构的更基本的理解来打破计算机代数几个问题的计算僵局。

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.