Complexity of Algebraic Structures

代数结构的复杂性

• 批准号: 0855601
• 负责人: Wolmer Vasconcelos
• 金额: $ 12.92万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0855601&HistoricalAwards=false
• 关键词: Complexity; Algebraic; Structures

The project introduces an approach to the study of theoretical aspects of certain classes of algebraic structures from the perspective of complexity. The main objective is the study of algebraic structures--a ring, an ideal, or even a module--as they undergo smoothing processes. These transformations enable them to support new constructions, including analytic ones. In the case of algebras, divisors acquire a group structure, cohomology tends to slim down, and it is an essential step in the desingularization of singular varieties. At its core is the inherent interest in those processes that add to the structure the solutions of collections of equations of integral dependence. Finding these equations, determining the properties of the assemblage of solutions and understanding the complexity costs of these tasks is a central region of research for commutative algebra. The assignments of measures of size, via multiplicity theories recently discovered by the proposer and his students, to the algebras and to the construction itself are key aspects of the project. As applications, the proposer seeks to predict how delicate techniques associated to smoothing processes will perform when applied to the solution of several problems of interest, and thereby suggest which mix of methods offer higher performance. They will also be employed to derive ordinary complexity counts for several of these problems without previously known classical counts.Commutative algebra, the subject area of the proposal, is foremost the study of systems of polynomial equations, and of its generalizations. It has elucidated several structures that occur among such systems, particularly those tagged as of Cohen-Macaulay type. These encode theoretical efficiencies in the derivation/prediction of its properties and offer superb computational economies. Often the full natural set of equations is not known at the outset so that methods and processes must be developed to find and analyze it. This proposal is focused on one central process, that of smoothing transformation. It will develop methods, grounded on the Cohen-Macaulay case, to predict properties of the closure, devise algorithms to find it and examine the limits of the behavior of arbitrary (even unknown) algorithms. The results and methods developed will be used for interaction where the subject meets algebraic geometry, combinatorics, geometric modeling, number theory and robotics.

该项目介绍了一种方法，从复杂性的角度研究某些类别的代数结构的理论方面。 主要目标是研究代数结构-环，理想，甚至是模-因为它们经历平滑过程。这些转换使它们能够支持新的结构，包括分析结构。在代数的情况下，因子获得了一个群结构，上同调趋于变细，这是奇异簇的去奇异化的一个重要步骤。在其核心是内在的兴趣，在这些过程中，添加到结构的解决方案的集合方程的整体依赖。找到这些方程，确定解决方案的集合的属性，并了解这些任务的复杂性成本是交换代数研究的中心领域。分配措施的大小，通过多重理论最近发现的提议者和他的学生，代数和建设本身是该项目的关键方面。作为应用程序，提议者试图预测如何微妙的技术相关联的平滑过程将执行时，应用到几个感兴趣的问题的解决方案，从而建议的方法组合提供更高的性能。 他们也将被用来获得普通的复杂性计数为这些问题没有以前已知的经典counts.Commutative代数，主题领域的建议，是最重要的研究系统的多项式方程，及其推广。它已经阐明了几个结构，发生在这样的系统，特别是那些标记为科恩-麦考利类型。这些编码的推导/预测其属性的理论效率，并提供卓越的计算经济。通常，一开始并不知道方程的全部自然集合，因此必须开发方法和过程来找到和分析它。 它将开发基于Cohen-Macaulay案例的方法来预测闭包的属性，设计算法来找到它并检查任意（甚至未知）算法的行为限制。 开发的结果和方法将用于互动的主题满足代数几何，组合数学，几何建模，数论和机器人。