RUI: FOUR PROBLEMS IN POLYTOPAL ALGEBRAIC COMBINATORICS

拟议的研究重点是离散几何，凸多面体，组合交换代数和代数K理论的十字路口的问题。涉及的主题是：正规多面体和有理锥的积分Caratheodory性质，仿射幺半群环的同调性质和K-理论性质，以及凸多面体范畴中的上纤维化。第一个研究方向提出了一个新的动力学方法正常多面体，而不是一个单一的多面体的静态图片的传统研究。这是通过在某个全局偏序集中编码正常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上的一些中心开放问题有很大的启发。第二个研究主题是一个算法的尝试，在反驳的猜想，仿射锥光滑投射环面品种是Koszul。这包括算法分析的几个密切相关的属性的独立利益：正常，二次生成，决议复曲面奇点等第三个研究课题是高K-理论的仿射幺半群环。奇异环的高K-理论的显式描述是一个罕见的现象。在这里，一个更好的多阶结构的所涉及的K-群是detertured，而不是较弱的分级结构已知的。第四个研究主题是关于凸多面体及其仿射映射的范畴。对如何将泛范畴概念应用于商多面体等假设对象提出了具体建议。组合数学是组织、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。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相关的计算。组合数学和抽象数学技术的相互作用，这是本研究的主旨，在过去的二十年中，在各种学科中产生了一些基本定理。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 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.