Algebraic geometry of toric varieties and its applications to combinatorics.

我们研究起源于环面品种几何的问题。环面变体是由凸多面体或扇形组合定义的几何对象。因此，环面品种的几何性质反映在多面体或扇形的组合中。环面变分理论在代数几何和组合学中都有应用。在代数几何中，环向变异体被认为比一般变异体简单得多，因此很难的猜想可以先在环向变异体上得到检验和证明。这种结果的例子首先证明了环面品种包括最小模型程序，镜像对称，和其他。****环面品种在组合学中也有应用。这通常涉及到从几何上解释多面体或扇形的一些组合特性。这方面最著名的例子是描述一个简单多面体在每个维度上可以有多少面。这些面数是用环变的上同调来描述的，由几何推导出的关系给出了这些面数的所有必要条件。***本建议的第一部分涉及多面体中点阵的组合。与此相对应的几何研究是环的考克斯环及其膨胀。提案的其余部分研究多面体的面数或面链。这在代数几何中对应于研究各种上同调理论，如交上同调、cd-索引、代数上同调理论等

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.**