RUI: Quantum, arithmetic, and categorial analysis of convex polytopes

RUI：凸多面体的量子、算术和分类分析

• 批准号: 1301487
• 负责人: Joseph Gubeladze
• 金额: $ 13.5万
• 依托单位: San Francisco State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301487&HistoricalAwards=false
• 关键词: RUI; Quantum; arithmetic; categorial; analysis

Lattice polytopes and convex polytopes form a natural habitat for a major part of the contemporary combinatorics. The first research direction, proposed in this project, involves a radically new idea of using analogies with quantum probabilistic physics. It introduces a new topological context for normal lattice polytopes. The second direction develops essentially new techniques for studying general convex polytopes. It is based on advanced categorial and homological machinery. The first line of research aims at shedding new light and making substantial progress on several outstanding open problems, such as unimodular triangulations and covers, Caratheodory rank, higher syzygies of the corresponding toric rings. The new idea of attacking these challenges draws from a physical interpretation of special discrete point configurations. Concrete suggestions in this direction to the PI were made by theoretical physicists and further connections to the physical context will be paid special attention. The second direction focuses on the category of convex polytopes and their affine maps. The goal is to devise a universal technique, providing a general context for various recent important constructions and results in polytope theory. The categorial approach leads to very concrete problems in classical polytopal combinatorics.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. Combinatorics of lattice polytopes 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, central to the proposed research, over the last decades has resulted in a number of fundamental theorems in a variety of disciplines. The proposal explores some of the deepest problems in polytopal combinatorics, employs ideas from a variety of disciplines and unexpected links between them, and offers concrete strategies for attacking central open questions. Applications range from algebraic geometry (the science of solution sets to systems of multidimensional polynomial equations) to integer programming/computer science, probability theory, and physics. The progress would have been unimaginable without computer assisted investigation and experimentation, the increased importance of which is related to the demand for algorithmic understanding of discrete structures. Both proposed research lines incorporate a strong - sometimes, crucial - computational element. Developing and implementing related algorithms is an excellent possibility for involving beginning graduate students.

格多面体和凸多面体构成了当代组合学的一个重要组成部分。在这个项目中提出的第一个研究方向涉及一个全新的想法，即使用量子概率物理学的类比。它为正规格多面体引入了一种新的拓扑上下文。第二个方向基本上是研究一般凸多面体的新技术。它是基于先进的范畴和同源机制。第一线的研究旨在摆脱新的光，并取得实质性进展的几个突出的开放问题，如单模三角剖分和覆盖，Caratheodory秩，较高syzygies相应的复曲面环。应对这些挑战的新想法来自对特殊离散点配置的物理解释。理论物理学家向PI提出了这个方向的具体建议，并将特别注意与物理背景的进一步联系。第二个方向主要研究凸多面体及其仿射映射。我们的目标是设计一个通用的技术，提供了一个一般的背景下，各种最近的重要建设和结果的多面体理论。范畴方法在经典的多面体组合学中导致非常具体的问题。组合学是组织、安排和分析离散数据的科学。一个说明性的例子是在平面中的凸多边形或在空间中的凸多面体中的整数点的集合。格多面体的组合学研究这样的点的配置编码在代数，几何和拓扑学的重要结构，而组合方法非常适合相关的计算。组合学和抽象数学技术的相互作用，中央提出的研究，在过去的几十年中，导致了一些基本定理在各种学科。该提案探讨了多面体组合学中一些最深层次的问题，采用了各种学科的思想和它们之间意想不到的联系，并提供了攻击中心开放问题的具体策略。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，概率论和物理学。如果没有计算机辅助的研究和实验，这些进展是不可想象的，其重要性的增加与对离散结构的算法理解的需求有关。这两个拟议的研究路线都包含了一个强大的-有时是关键的-计算元素。开发和实现相关的算法是一个很好的可能性，涉及开始研究生。