Topology of Algebraic Varieties and Enumerative Combinatorial Geometry

代数簇拓扑与枚举组合几何

基本信息

批准号：
1701305
负责人：
Botong Wang
金额：
$ 9.47万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701305&HistoricalAwards=false
关键词：
Topology Algebraic Varieties Enumerative Combinatorial

项目摘要

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.

列举的组合几何学研究基本计数几何组合对象的问题。这样一个问题的一个例子是确定平面中给定数量的线数之间可能的相交点数。该项目使用代数几何形状的方法研究了此类组合问题。某些组合不变性已显示具有代数几何性质，也就是说，存在与特殊组合物体相关的代数品种，并且可以使用相应的代数品种表达组合不变性。一旦实现了这种代数的几何性质，就可以将组合问题转化为有关代数品种的问题，并有可能将其减少到代数几何形状中的已知结果。组合物体下面的基础代数几何结构也会导致对组合对象的更深入的了解。矩形理论是组合几何学的基本主题。该项目的第一个目标是探索在某些领域可以实现的矩形数值不变性的代数几何性质。更确切地说，该项目旨在以某些相关代数差异的（典型或奇异）的共同体戒指来表达数值不变性，从而获得了矩阵的新特性。更雄心勃勃的目标是在代数几何形状中开发定理的组合概括，并为不可通知的矩阵提供特性。