RUI: FOUR PROBLEMS IN POLYTOPAL ALGEBRAIC COMBINATORICS

所提出的研究集中在离散几何、凸多面体、组合交换代数和代数K-理论的十字路口问题。所涉及的主题包括：正规多面体和有理锥的积分Caratheodory性质，仿射么半群环的同调性质和K-理论性质，以及凸多面体范畴中的余贝塔。第一个研究方向提出了一种新的动态方法来研究正常多面体，而不是传统的研究单个多面体的静态图像。这是通过编码某个全局偏序集中正常多面体的相互作用来完成的。相关的拓扑学有可能揭示一些关于Hilbert基的中心公开问题。第二个研究主题是反驳关于光滑射影环簇上的仿射锥是Koszul的猜想的算法尝试。这包括几个与独立兴趣密切相关的性质的算法分析：正规性、二次生成、环状奇点的分解等。第三个研究课题是仿射么半环的高K-理论。对奇异环的高K-理论进行显式刻画是一种罕见的现象。与目前已知的较弱的分次结构不同，这里猜想了所涉及的K-群的一个更精细的多分次结构。第四个研究主题是关于凸多面体及其仿射映射的范畴。对如何将普遍范畴概念应用于商多面体等假设对象提出了具体建议。组合学是一门组织、排列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，这是这项研究的主旨，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。后一方面使得该项目特别适合于吸引初学研究生参与研究。

The proposed research focuses on problems at the crossroads of discrete geometry, convex polytopes, combinatorial commutative algebra, and algebraic K-theory. The topics involved are: integral Caratheodory property of normal polytopes and rational cones, homological and K-theoretical properties of affine monoid rings, and cofibrations in the category of convex polytopes. The first research direction proposes a new dynamical approach to normal polytopes, as opposed to the traditional study of the static picture of a single polytope. This is done via encoding the interactions of normal polytopes in a certain global poset. The associated topology has a potential of shedding much light to some central open questions on Hilbert bases. The second research topic is an algorithmic attempt at disproving the conjecture that the affine cones over smooth projective toric varieties are Koszul. This includes algorithmic analysis of several closely related properties of independent interest: normality, quadratic generation, resolutions of toric singularities etc. The third research topic is higher K-theory of affine monoid rings. An explicit description of higher K-theory of a singular ring is a rare phenomenon. Here a finer multigraded structure of the involved K-groups is conjectured, as opposed to the weaker graded structures known so far. The fourth research topic concerns the category of convex polytopes and their affine maps. Concrete suggestions are made on how to apply universal categorial concepts to such hypothetical objects as quotient polytopes.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 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, 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.