RUI: Convex Point Configurations in Algebraic Combinatorics

RUI：代数组合中的凸点配置

• 批准号: 0600929
• 负责人: Joseph Gubeladze
• 金额: $ 10.54万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600929&HistoricalAwards=false
• 关键词: RUI; Convex; Point; Configurations; Algebraic

The proposed research lies at the crossroads of discrete geometry, combinatorial commutative algebra, toric geometry, higher algebraic K-theory of rings, and the theory of convex polytopes. The expected results will have applications to other fields, such as geometric number theory, integer programming (via Groebner bases of binomial ideals) and, potentially, physical sciences (algebraic structure of physical units). The first group of problems concerns the Hilbert bases of polyhedral cones - distinguished lattice point configurations in discrete geometry for which no satisfactory geometric characterization is known to date. Very concrete working conjectures are made on Caratheodory ranks in higher dimensions, an effective uniform version of Knudsen-Mumford's result on unimodular triangulations, and point configurations with extremal arithmetic properties. Despite much effort in recent years the current state of the art is that positive results are very difficult to obtain as the dimension of the space increases, and in higher dimensions only a few striking counterexamples are known. Verification of the proposed conjectures would shed much light on the global picture. The second group of problems is more related to algebraic geometry. The problems concern sophisticated homological and high K-theoretical aspects of lattice point configurations, such as Koszul property of projective embeddings of smooth toric varieties and K-homotopy invariance of toric singularities. Related algorithmic issues include analysis of smooth polytopes and factorization of invertible matrices over monomial algebras. The project also aims at a deeper understanding of the category of polytopes (mapping, tensor and cofiber objects).Combinatorics is the science of organizing, arranging and analyzing discrete data. An illustrative example is the set of integer points in a convex polygon in the plane or in a convex polytope in the space. Algebraic combinatorics studies such point configurations to encode important constructions in algebra, geometry, and topology, while combinatorial methods are well suited for related computations. The interaction of combinatorics and abstract mathematical techniques (which is the leitmotif of this research) over the last two decades has resulted in a number of fundamental theorems in a variety of disciplines. Applications range from algebraic geometry (the science of solution sets to systems of multidimensional polynomial equations) to integer programming, computer science, probability theory, physics, cryptography etc. The progress would have been unimaginable without computer assisted investigation and experimentation, the increasing importance of which is related to the demand for explicit or algorithmic understanding of discrete structures. The latter aspect makes the project especially well suited for engaging beginning graduate students in the research.

所提出的研究处于离散几何、组合交换代数、环面几何、环的高等代数K理论和凸多面体理论的十字路口。预期结果将应用于其他领域，例如几何数论、整数规划（通过二项式理想的 Groebner 基）以及潜在的物理科学（物理单位的代数结构）。第一组问题涉及多面锥体的希尔伯特基 - 离散几何中的独特晶格点配置，迄今为止尚无令人满意的几何特征。非常具体的工作猜想是在更高维度的 Caratheodory 等级上进行的，这是 Knudsen-Mumford 在单模三角剖分上的结果的有效统一版本，以及具有极值算术属性的点配置。尽管近年来付出了很多努力，但目前的技术水平是，随着空间尺寸的增加，很难获得积极的结果，并且在更高的尺寸中，仅已知一些引人注目的反例。对所提出的猜想的验证将为全球情况提供更多线索。第二组问题与代数几何更相关。这些问题涉及晶格点配置的复杂同调和高 K 理论方面，例如平滑环面簇射影嵌入的 Koszul 性质和环面奇点的 K 同伦不变性。相关的算法问题包括平滑多面体的分析和单项代数上可逆矩阵的因式分解。该项目还旨在更深入地了解多面体类别（映射、张量和共纤维对象）。组合学是组织、排列和分析离散数据的科学。说明性示例是平面中的凸多边形或空间中的凸多面体中的整数点的集合。代数组合学研究此类点配置，以编码代数、几何和拓扑中的重要结构，而组合方法非常适合相关计算。在过去的二十年中，组合数学和抽象数学技术（这是本研究的主题）的相互作用在各个学科中产生了许多基本定理。应用范围从代数几何（多维多项式方程组的解集科学）到整数规划、计算机科学、概率论、物理学、密码学等。如果没有计算机辅助研究和实验，这些进步将是难以想象的，这与对离散结构的显式或算法理解的需求有关。后一个方面使得该项目特别适合让刚起步的研究生参与研究。