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

本文的研究方向是代数几何，并将其应用于奇点理论，采用代数拓扑方法。它的目的是扩展先前提出的PDRA在全局奇点理论中的研究，该理论涉及某些非约代数群作为微分同构群发生的作用。该项目的目标是扩展这些想法，利用PI和她的合作者Doran最近和当前的研究成果，构建代数几何中非约化群行为的商空间的一般理论。代数几何将抽象代数的技巧与几何的语言和直觉相结合。它在现代数学中占据中心位置，并且与物理学有多种联系，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多项式方程：代数几何学者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的作用，特别是代数拓扑中的局部化方法。拓扑学背后的鼓舞人心的见解是，许多几何问题的答案并不依赖于所涉及物体的精确形状，而是依赖于一个更宽松的形状概念；将代数几何的优良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：即群体行动。对称性在许多数学和物理中，特别是在代数几何和拓扑学中，都是非常重要的。群行动的不动点集合通常存储有关空间拓扑的重要信息；这是本项目中用于研究代数几何中商空间拓扑的局部化定理的基础。模空间（几何对象族的参数空间）是代数几何的核心问题之一，在几何和理论物理的相关领域具有重要意义，而商空间往往是构造和理解模空间的基础。全局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，为了理解全局映射，我们研究欧几里得空间之间的局部映射，但必须考虑坐标的变化。因此，对高度复杂的、无限维的、不可约的局部微分同构群进行认识，并根据它们的作用取适当的商是十分重要的。对于一类重要的奇点，即Morin奇点，本文提出的PDRA构造了一个多重不变量的迭代残差公式。本项目旨在利用局部化方法寻找更一般的非约商的相似迭代残数公式，并将其应用于比Morin奇点更一般的全局奇点理论中。

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.