RUI: Convex Point Configurations in Algebraic Combinatorics

拟议的研究在于离散几何，组合交换代数，环面几何，高等代数K-理论的环，和凸多面体的理论的十字路口。预期的结果将应用于其他领域，如几何数论，整数规划（通过Groebner基地的二项式理想），并可能，物理科学（代数结构的物理单位）。第一组问题涉及的希尔伯特基地的多面体锥-杰出的格点配置在离散几何没有令人满意的几何表征是已知的日期。非常具体的工作成果是Caratheodory等级在更高的维度，一个有效的统一版本的努森-芒福德的结果unimodular三角剖分，和点配置与极值算术性能。尽管近年来做了很多努力，但随着空间维数的增加，很难得到肯定的结果，而且在更高的维数下，只有少数引人注目的反例是已知的。核查拟议的证据将有助于了解全球情况。第二组问题与代数几何关系更密切。问题涉及复杂的同调和高K-理论方面的格点配置，如Koszul性质的光滑环面品种和K-同伦不变性的环面奇点的投射嵌入。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分解。该项目还旨在更深入地了解多面体的类别（映射，张量和cofiber对象）。组合数学是组织，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。代数组合学研究这样的点的配置来编码代数，几何和拓扑学中的重要结构，而组合方法非常适合相关的计算。在过去的二十年里，组合学和抽象数学技术（这是本研究的主题）的相互作用导致了各种学科中的一些基本定理。应用范围从代数几何（科学的解决方案集系统的多维多项式方程），以整数规划，计算机科学，概率论，物理学，密码学等的进展本来是不可想象的，如果没有计算机辅助调查和实验，日益重要的是有关的需求明确或算法的理解离散结构。后一个方面使该项目特别适合从事研究生的研究。

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.