RUI: Convex Point Configurations in Algebraic Combinatorics

所提出的研究位于离散几何、组合交换代数、环面几何、环的高代数K-理论和凸多面体理论的十字路口。预期的结果将应用于其他领域，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体锥体的Hilbert基-离散几何中可区分的格点配置，迄今尚无令人满意的几何特征。在高维Caratheodory秩上提出了非常具体的工作猜想，它是Knudsen-Mumford关于单模三角剖分结果的有效统一版本，以及具有极值算术性质的点配置。尽管近年来做了很多努力，但目前的技术状况是，随着空间维度的增加，很难获得积极的结果，并且在更高的维度中，只有少数引人注目的反例是已知的。对提出的猜想的验证将对全球情况有很大的帮助。第二组问题更多地与代数几何有关。这些问题涉及到格点构型的复杂的同调和高K理论方面，如光滑环面簇的射影嵌入的Koszul性质和环面奇点的K-同伦不变性。相关的算法问题包括光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(映射、张量和共纤维对象)的类别。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平面中的凸多边形或空间中的凸多面体中的整点集。代数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。在过去的二十年里，组合学和抽象数学技术的相互作用(这是本研究的主题)在不同的学科中产生了许多基本定理。应用范围从代数几何(解集的科学到多维多项式方程组)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的调查和实验，这一进步是不可想象的，这与对离散结构的显式或算法理解的需求有关。后一方面使该项目特别适合于吸引初学研究生参与研究。

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.