Homology of Monomial and Toric Ideals

这个项目关注的是与单项式理想和环面品种相关的自由分辨率。它是交换代数、代数几何和组合学的交叉点。该项目的目标是在一些特殊情况下构建最小自由分辨率，构造非最小结构化分辨率，并获得Betti数的界。其中一些问题与格罗布纳基理论和整数规划密切相关。这是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，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

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.