Stable and unstable cohomology of moduli spaces

模空间的稳定和不稳定上同调

基本信息

批准号：
EP/M027783/1
负责人：
Oscar Randal-Williams
金额：
$ 11.58万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM027783%2F1
关键词：
Stable unstable cohomology moduli spaces

项目摘要

In classical mathematics, mathematical objects and their properties are usually considered one at a time: we might consider a triangle in the plane, with its associated lengths and angles, and ask questions about it, such as what is its perimeter, or area. In the 20th Century it became increasingly understood that it can be profitable to consider the collection of all mathematical objects of some type: we might consider the space whose points correspond to triangles in the plane, in which moving the three vertices of the triangle around defines a path.These spaces of mathematical objects, "moduli spaces" as they are known, have become an object of study which can be approached from many areas of mathematics, each of which give a particular insight. The most intensely studied moduli space, and the first example of one, is the moduli space of Riemann surfaces. This is difficult to visualise directly: a point of this space corresponds to a surface, such as a ball or the layer of sugar on a (American) doughnut, and moving around in this space corresponds to bending and stretching the surface. Because this space is so difficult to visualise, abstract tools must be used to get a feel for it: to get an idea of the topological complexity of the space, the most successful of these are homology and cohomology.This project will investigate moduli spaces of higher-dimensional manifolds, focussing on their homology and cohomology. That is, it will consider spaces whose points are d-dimensional manifolds (so rather than being surfaces, which locally look like 2-dimensional space, they are spaces which locally look like d-dimensional space), and where movement in this space corresponds to bending and stretching. Manifolds are the fundamental objects studied in Geometry, so the space of all manifolds of a given dimension is intimately related to many questions that can be asked in this subject. Part of this project is to use and develop a strong new tool which has been created by Galatius and the PI, in order to investigate geometric questions in a certain ``stable range". In addition, the project will introduce new methods to understand moduli spaces of manifolds outside of this ``stable range", where a systematic picture is lacking.

在经典数学中，通常一次将数学对象及其属性视为一个：我们可能会考虑平面中的三角形，其长度和角度，并提出有关它的问题，例如其周长或区域是什么。在20世纪，越来越了解，考虑某种类型的所有数学对象的收集是可以有利可图的：我们可以考虑该空间与飞机上的三角形相对应的空间，在该空间中，三角形的三个顶点定义了一条路径。洞察力。最深入研究的模量空间，第一个示例是Riemann表面的模量空间。这很难直接可视化：该空间的一个点对应于表面，例如球或（美国）甜甜圈上的糖层，并且在这个空间中移动对应于弯曲和拉伸表面。由于这个空间很难可视化，因此必须使用抽象工具来了解它：要了解空间的拓扑复杂性，其中最成功的是同源性和共同体学。该项目将研究高维歧管的模数空间，专注于其同源和同源性和同胞。也就是说，它将考虑其点为d维歧管的空间（因此，它们在局部看起来像二维空间，而不是表面，它们是本地看起来像D维空间的空间），并且该空间中的运动在该空间中对应于弯曲和伸展。歧管是几何形状研究的基本对象，因此给定维度的所有歧管的空间与许多问题中的许多问题密切相关，这些问题可以在此主题中提出。该项目的一部分是使用和开发一个强大的新工具，该工具是由Galatius和Pi创建的，以调查某个``稳定范围''的几何问题。此外，该项目将引入新的方法来了解``稳定范围''之外的流形的模量空间。