Topology of Algebraic Varieties and Enumerative Combinatorial Geometry

枚举组合几何研究几何组合对象的基本计数问题。这样一个问题的一个例子是确定平面上给定数量的直线之间可能的交点数量。这个项目用代数几何的方法来研究这样的组合问题。一些组合不变量已被证明具有代数几何性质，即存在与特定组合对象相关联的代数变量，并且可以用相应的代数变量来表示组合不变量。一旦认识到这样的代数几何性质，人们就可以将组合问题转化为关于代数变异的问题，并有可能将它们简化为代数几何中的已知结果。组合对象下面的代数几何结构也可以导致对组合对象的更深层次的理解。矩阵理论是组合几何中的一门基础学科。本项目的第一个目标是探索在某些域上可实现的拟阵数值不变量的代数几何性质。更确切地说，该项目旨在用一些相关代数变种的（可变或奇异）上同环来表示数值不变量，从而获得拟阵的新性质。更远大的目标是发展代数几何定理的组合推广，并推导出不可实现的拟阵的性质。

Enumerative combinatorial geometry studies fundamental counting questions for geometric combinatorial objects. One example of such a question is to determine the possible number of intersection points among a given number of lines in the plane. This project investigates such combinatorial questions using methods from algebraic geometry. Some combinatorial invariants have been shown to have algebraic geometric nature, that is, there are algebraic varieties associated to special combinatorial objects, and it is possible to express combinatorial invariants using the corresponding algebraic varieties. Once such algebraic geometric nature is realized, one can translate combinatorial questions into questions about algebraic varieties, with the potential to reduce them to known results in algebraic geometry. The underlying algebraic geometry structures beneath combinatorial objects can also lead to deeper understanding of the combinatorial objects. Matroid theory is a fundamental subject in combinatorial geometry. The first goal of this project is to explore the algebraic geometric nature of the numerical invariants of matroids that are realizable over some field. More precisely, the project aims to express the numerical invariants in terms of the (étale or singular) cohomology ring of some associated algebraic varieties, obtaining new properties of the matroids. The more ambitious goal is to develop combinatorial generalizations of theorems in algebraic geometry and to derive properties for non-realizable matroids.