Homology of Monomial and Toric Ideals

Kleiman 9700564这个项目关注与单项理想和环面簇相关的自由归结。 它处于交换代数、代数几何和组合数学之间的界面。 该项目的目标是在某些特殊情况下构造极小自由归结，构造非极小结构归结，并获得Betti数的界。 有些问题与Groebner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。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。 如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，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.