Groebner Fans in Combinatorics, Representation Theory and Commutative Algebra: Theory and Computation

托马斯的DMS-0401047获奖摘要：Groebner基本理论是当今代数几何和交换代数中的核心理论和计算工具，应用于计算机科学、优化、机器人技术、统计学和计算机建模等众多领域。理想或模的Groebner扇是引入Groebner基的权向量空间的分解，使得扇中的每个锥索引不同的Groebner基。这个项目描述了围绕Groebner风扇的组合学、最优化、表示论和代数几何四个问题。第一个项目研究复曲面理想的有限理想的复杂性、几何和同调性质。这些问题的答案haveimplications在整数规划和计算toricGroebner基地本身有几个应用。其次考察了Kalai的单纯复形代数移位理论中权向量的变化。 第三个项目研究的多面体几何的模量的表现的McKay的（部分）解决奇异性所产生的广义McKay对应。最后一个目标是开发一个计算任意多项式理想的Groebner扇的软件包。 应用包括热带簇的计算和轨道闭合的研究和退化的表示有限维代数。拟议的工作将采用组合学，离散几何和优化的工具来解决问题，从交换代数，表示论和代数几何。 所有项目都有一个强大的计算组件，旨在有效的算法，并作为一个实验室，揭示新的结构定理。这些合作包括三名研究生和三名现任或近期的博士后。

Abstract for award DMS-0401047 of Thomas: Groebner basis theory is today a central theoretical and computationaltool in algebraic geometry and commutative algebra with applicationsin myriad fields such as computer science, optimization, robotics,statistics, and computer modeling. The Groebner fan of an ideal ormodule is the decomposition of the space of weight vectors that induceGroebner base, such that each cone in the fan indexes a distinctGroebner basis. This project describes four problems fromcombinatorics, optimization, representation theory and algebraicgeometry that revolve around Groebner fans. The first projectinvestigates the complexity, geometry and homological properties ofinitial ideals of toric ideals. Answers to these questions haveimplications in integer programming and on the computation of toricGroebner bases which themselves have several applications. The next examines the variation of weight vectors in Kalai's theory ofalgebraic shifting for simplicial complexes. The third projectstudies the polyhedral geometry of moduli of resentations of the McKayquiver which (partially) resolve singularities that arise ingeneralized McKay correspondence. The last goal is to develop asoftware package for computing Groebner fans of arbitrary polynomialideals. Applications include the computation of tropical varieties andthe study of orbit closures and degenerations of representations offinite dimensional algebras.The proposed work will employ tools from combinatorics, discretegeometry and optimization to solve problems from commutative algebra,representation theory and algebraic geometry. All projects have astrong computational component that aims at effective algorithms andserves as a laboratory for uncovering new structure theorems. Thecollaborations include three graduate students and three current orrecent postdocs.