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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与奇异性理论的应用，并使用代数拓扑的方法。它的目的是扩展早期的研究所提出的PDRA在全球奇异性理论，其中涉及的行动，某些非约化代数群发生的非同态群。该项目的目标是扩展这些想法，利用PI和她的合作者Doran最近和当前的研究，为代数几何中的非约化群作用构建商空间的一般理论。代数几何结合了抽象代数的技术和几何的语言和直觉。它在现代数学中占据中心地位，并且与物理学有着多种联系，例如通过规范理论和弦理论。代数几何的中心对象是多变量的多项式方程：代数几何学家试图理解这样一个方程组的全部解。拓扑学在这个项目中也起着关键作用，特别是代数拓扑中的局部化方法。拓扑学背后的启发性见解是，许多几何问题的答案并不依赖于所涉及对象的精确形状，而是依赖于一个更松散的形状概念;将代数几何的精细工具与拓扑方法相结合，产生了许多重要的结果。这个项目中剩下的关键因素是对称性：也就是群体行动。对称性在许多数学和物理学中具有根本的重要性，特别是在代数几何和拓扑学中。一个群作用的不动点集通常存储了关于空间拓扑的重要信息;这是本项目中为了研究代数几何中商空间的拓扑而使用的局部化定理的基础。商空间通常是构造和理解模空间（几何对象族的参数空间）的基础，模空间是代数几何的中心问题之一，在几何和理论物理的相关领域也非常重要。整体奇点理论的主要研究对象是流形之间的映射。在奇点理论中，为了理解整体映射，我们研究欧氏空间之间的局部映射，但必须考虑坐标的变化。因此，理解局部自同构群是非常复杂的、无限维的、非约化的群，并通过它们的作用取适当的自同构是很重要的。拟议的PDRA已经构建了一个迭代剩余公式，某些相关的不变量称为multidegrees，一类重要的奇异性称为Morin奇点。这个项目的目的是使用本地化的方法来找到类似的迭代剩余公式更一般的非约化的等价物，并将它们应用于全球奇点理论更一般的情况下，比莫林奇点。