RUI: FOUR PROBLEMS IN POLYTOPAL ALGEBRAIC COMBINATORICS

提出的研究集中在离散几何，凸多边形，组合交换代数和代数k理论的十字路口的问题。所涉及的主题是：正多边形和有理锥的积分卡拉多性，仿射一元环的同调和k -理论性质，以及凸多面体范畴的振动。第一个研究方向提出了一种新的动态方法来研究正多面体，而不是传统的研究单个多面体的静态图像。这是通过编码一个特定的全局偏序集的正多面体的相互作用来完成的。相关的拓扑结构有可能对希尔伯特基础上的一些核心开放性问题提供更多的启示。第二个研究课题是用一种算法来反驳光滑射影环上的仿射锥是Koszul的猜想。这包括对几个密切相关的独立性质的算法分析：正态性，二次生成，环奇点的解决等。第三个研究课题是仿射一元环的高k理论。对奇异环的高k理论的明确描述是一种罕见的现象。在这里，与目前已知的较弱的梯度结构相反，所涉及的k群的更精细的多梯度结构被推测出来。第四个研究课题是凸多面体的范畴及其仿射映射。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上的凸多边形或空间上的凸多边形中的整数点的集合。格多面体的代数组合学研究这些点的构型来编码代数、几何和拓扑中的重要结构，而组合方法则非常适合于相关的计算。组合学和抽象数学技术的相互作用，这是本研究的主题，在过去的二十年里，在各种学科中产生了许多基本定理。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、物理学、密码学等。如果没有计算机辅助的调查和实验，这些进展将是不可想象的，这与对离散结构的明确或算法理解的需求有关。后一个方面使得这个项目特别适合让刚开始研究的研究生参与。

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.