Algebraic geometry of toric varieties and its applications to combinatorics.

我们研究的问题起源于几何的复曲面品种。复曲面变体是由凸多面体或扇形组合定义的几何对象。复曲面簇的几何性质因此反映在多面体或扇形的组合学中。* 复曲面簇的理论在代数几何和组合数学中都有应用。在代数几何中，复曲面簇被认为比一般簇简单得多，所以硬拓扑可以首先在复曲面簇上得到检验和证明。这些结果的例子首先证明复曲面品种包括最小模型程序，镜像对称，和其他人。复曲面的变种在组合数学中也有应用。这些通常涉及解释一些组合性质的多面体或扇形几何。这方面最著名的例子是描述一个简单的多面体可以有多少个面。这些面数是用复曲面簇的上同调来描述的，来自几何的关系给出了这些数的所有必要条件。* 本提案的第一部分涉及多面体中格点的组合学。几何对应这是研究考克斯环的环面品种和他们的爆破。建议的其余部分研究多面体的面或面链的数量。这在代数几何中对应于学习各种上同调理论，如交上同调，CD-指数，代数配边理论等。

We study problems originating in the geometry of toric varieties. Toric varieties are geometric objects that are defined combinatorially by a convex polytope or a fan. The geometric properties of a toric variety are therefore reflected in the combinatorics of the polytope or fan. ***The theory of toric varieties has applications in both algebraic geometry and combinatorics. In algebraic geometry toric varieties are considered as being much simpler than general varieties, so hard conjectures can first be tested and proved on toric varieties. Examples of such results first proved for toric varieties include the minimal model program, mirror symmetry, and others.****Toric varieties also have applications in combinatorics. These typically involve interpreting some combinatorial property of a polytope or a fan geometrically. The most well-known example of this is the description of how many faces of each dimension a simple polytope can have. These face numbers are described in terms of cohomology of the toric variety, and the relations coming from geometry give all the necessary conditions on these numbers. ***The first part of this proposal relates to the combinatorics of lattice points in a polytope. The geometric counterpart of this is the study of Cox rings of toric varieties and their blowups. The remaining parts of the proposal study the number of faces or chains of faces of a polytope. This in algebraic geometry corresponds to studying various cohomology theories, such as intersection cohomology, cd-index, algebraic cobordism theory and others.**