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

晶格多面体和凸多面体构成了当代组合数学的主要部分的自然栖息地。该项目提出的第一个研究方向涉及一种全新的想法，即使用量子概率物理学的类比。它为正规晶格多面体引入了一种新的拓扑学背景。第二个方向本质上是研究一般凸多面体的新技术。它基于先进的范畴和同源机制。第一条线的研究旨在揭示和解决一些悬而未决的问题，如单模三角剖分和覆盖、Caratheodory阶数、相应环面环的高阶合子等。应对这些挑战的新想法源于对特殊离散点配置的物理解释。理论物理学家在这方面向PI提出了具体建议，并将特别注意与物理环境的进一步联系。第二个方向是关于凸多面体及其仿射映射的范畴。其目标是设计一种通用的技术，为多面体理论中各种最近的重要结构和结果提供一个普遍的背景。范畴方法导致了经典多面体组合学中非常具体的问题。组合学是一门组织、安排和分析离散数据的科学。一个说明性的例子是平面中的凸多边形或空间中的凸多面体中的整点集。格多面体的组合学研究这样的点配置来编码代数、几何和拓扑中的重要结构，而组合方法非常适合于相关的计算。在过去的几十年里，作为拟议研究的核心的组合学和抽象数学技术的相互作用，在不同的学科中产生了许多基本定理。该提案探讨了多面体组合数学中一些最深层次的问题，采用了来自不同学科的想法和它们之间意想不到的联系，并提供了解决中心开放问题的具体策略。应用范围从代数几何(解集的科学到多维多项式方程系统)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调查和实验，这一进展将是不可想象的，其重要性的增加与对离散结构的算法理解的需求有关。这两条拟议的研究路线都包含了一个强大的--有时是至关重要的--计算元素。开发和实施相关的算法是一个很好的可能性，让初学研究生参与进来。

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.