Mathematical Sciences: Commutative Algebra, Algebraic Geometry, and Computational Issues

数学科学：交换代数、代数几何和计算问题

• 批准号: 9201468
• 负责人: Ira Gessel
• 金额: $ 31.12万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201468&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Commutative; Algebra; Algebraic

This research is concerned with problems in commutative algebra, algebraic geometry and computational issues in those fields. In commutative algebra, the principal investigator will study linear resolutions and Huneke's question on the structure of free resolutions. In algebraic geometry, he will consider a generalized Castelnuovo conjecture, the family of compactified Jacobians by projective stability, degeneracy of secant and tangent varieties by methods from differential geometry, as well as ribbons, K3 carpets and supermanifolds. The computational parts of this project are concerned with criteria for systems of parameters and development of the computer algebra system, Macaulay. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields, as well as, in theoretical computer science and robotics.

本研究涉及交换代数、代数几何中的问题以及这些领域中的计算问题。在交换代数中，主要研究人员将研究线性归结和Huneke关于自由归结结构的问题。在代数几何中，他将考虑广义Castelnuovo猜想，通过射影稳定性压缩的雅可比家族，通过微分几何方法的割线和切变簇的简并性，以及带状、K3地毯和超流形。这个项目的计算部分涉及参数系统的标准和计算机代数系统Macaulay的发展。这是对代数几何领域的研究，代数几何是现代数学中最古老的部分之一，但在过去的25年里，它取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及理论计算机科学和机器人学中也找到了应用。