Localisation on quotients by non-reductive group actions and global singularity theory

非还原群作用和全局奇点理论对商的局部化

基本信息

批准号：
EP/G000174/1
负责人：
Frances Kirwan
金额：
$ 30.37万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG000174%2F1
关键词：
Localisation quotients non reductive group

项目摘要

The proposed research lies in algebraic geometry with applications in singularity theory, and uses methods of algebraic topology. It aims to extend earlier research by the proposed PDRA in global singularity theory, which involves actions of certain non-reductive algebraic groups which occur as diffeomorphism groups. The goal of the proposed project is to extend these ideas, using recent and current research by the PI and her collaborator Doran towards a general theory for constructing quotient spaces for non-reductive group actions in algebraic geometry.Algebraic geometry combines techniques of abstract algebra with the language and intuition of geometry. It occupies a central place in modern mathematics and also has multiple connections with physics, for example through gauge theory and string theory. The central objects of algebraic geometry are polynomial equations in many variables: algebraic geometers attempt to understand the totality of the solutions of such a system of equations. Topology also plays a key role in this project, especially localisation methods in algebraic topology. The motivating insight behind topology is that answers to many geometric problems depend not on the precise shape of the objects involved, but rather on a much looser concept of shape; combining the fine tools of algebraic geometry with topological approaches has resulted in many important results. The remaining crucial ingredient in this project is symmetry: that is, group actions. Symmetries are of fundamental importance throughout much of mathematics and physics, in particular in algebraic geometry and topology. The set of fixed points of a group action often stores significant information about the topology of a space; this is the basis for the localisation theorems to be used in this project in order to study the topology of quotient spaces in algebraic geometry. Quotient spaces are often fundamental in the construction and understanding of moduli spaces (parameter spaces for families of geometric objects), which is one of the central problems of algebraic geometry, and is of great importance in related areas of geometry and of theoretical physics.The main objects of study in global singularity theory are maps between manifolds. In singularity theory, in order to understand global maps, we study local maps between Euclidean spaces, but it is necessary to take account of changes of coordinates. Thus it is important to understand the local diffeomorphism groups, which are highly complicated, infinite-dimensional, non-reductive groups, and to take appropriate quotients by their actions. The proposed PDRA has constructed an iterated residue formula for certain associated invariants called multidegrees, for an important class of singularities called Morin singularities. This project aims to use localisation methods to find similar iterated residue formulas for much more general non-reductive quotients, and to apply them in global singularity theory to more general situations than that of Morin singularities.

拟议的研究在于代数几何形状，其在奇异理论中的应用，并使用代数拓扑的方法。它的目的是扩展拟议的PDRA在全球奇异理论中的早期研究，该研究涉及某些非降级代数群体作为差异群体的行为。拟议项目的目的是利用PI及其合作者Doran的最新研究和当前的研究来扩展这些想法，以构建代数几何形状中非还原群体行动的商空间的一般理论。代数几何结合了抽象代数的技术与几何学的语言和直觉。它在现代数学中占据着核心地位，并且与物理学有多种联系，例如通过量规理论和弦理论。代数几何形状的中心对象是许多变量中的多项式方程：代数几何器试图理解这种方程式系统的总体。拓扑在该项目中也起着关键作用，尤其是代数拓扑中的本地化方法。拓扑背后的激励见解是，对许多几何问题的答案并不取决于所涉及的对象的精确形状，而是基于形状的更宽松的概念。将代数几何形状的精细工具与拓扑方法相结合，从而产生了许多重要的结果。该项目中其余的至关重要成分是对称性：即小组动作。对称性在许多数学和物理学，尤其是代数几何和拓扑中至关重要。小组行动的固定点集通常会存储有关空间拓扑的重要信息。这是在该项目中使用本地化定理的基础，以研究代数几何形状中商空间的拓扑。商空间通常是对模量空间的构建和理解（几何对象家族的参数空间）的基础，这是代数几何形状的核心问题之一，并且在基本物理学的相关领域和理论物理学的相关领域至关重要。全球奇异理论中的主要物理学的主要物理学是示范的。在奇异理论中，为了理解全球地图，我们研究了欧几里得空间之间的局部图，但是有必要考虑坐标的变化。因此，重要的是要了解高度复杂，无限二维，非还原群的局部差异群体，并通过其行为来采取适当的人。拟议的PDRA已为某些称为多边形的相关不变式构建了一个迭代的残基公式，用于一种称为Morin奇点的重要类别。该项目旨在使用本地化方法来找到类似的迭代残基公式，以实现更一般的非还原词，并将其应用于全球奇异理论中，而不是莫林奇异性。