Homology of Monomial and Toric Ideals

单项式和环面理想的同调

• 批准号: 9700564
• 负责人: Steven Kleiman
• 金额: $ 5.47万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700564&HistoricalAwards=false
• 关键词: Homology; Monomial; Toric; Ideals

Kleiman 9700564 This project is concerned with free resolutions related to monomial ideals and toric varieties. It is in the interface between commutative algebra, algebraic geometry and combinatorics. The goal of the project is to build minimal free resolutions in some special cases, construct non-minimal structured resolutions, and obtain bounds on Betti numbers. Some of the problems are closely related to Groebner basis theory and integer programming. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, 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 physics, theoretical computer science, and robotics.

Kleiman 9700564这个项目涉及与单项理想和环面簇有关的自由分解。它处于交换代数、代数几何和组合学的交界处。该项目的目标是在某些特殊情况下构造最小自由归结，构造非最小结构化归结，并获得Betti数的界。其中一些问题与Groebner基理论和整数规划密切相关。这是在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，它已经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，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。