Stable and unstable cohomology of moduli spaces

在经典数学中，数学对象及其性质通常一次只考虑一个：我们可以考虑平面上的一个三角形，以及它相关的长度和角度，并询问有关它的问题，例如它的周长或面积是多少。在20世纪，人们越来越认识到，考虑某种类型的所有数学对象的集合是有益的：我们可以考虑其点对应于平面上的三角形的空间，在该空间中，绕着三角形的三个顶点移动定义了一条路径。这些数学对象的空间，也就是他们所知的“模空间”，已经成为一个可以从许多数学领域来探讨的研究对象，每个领域都给出了特定的见解。研究最深入的模空间，也是其中的第一个例子，是黎曼曲面的模空间。这很难直接可视化：这个空间的一个点对应于一个表面，例如(美国)甜甜圈上的一个球或糖层，在这个空间中四处移动对应着弯曲和拉伸表面。因为这个空间很难可视化，所以必须使用抽象工具来感受它：为了了解空间的拓扑复杂性，最成功的工具是同调和上同调。本项目将研究高维流形的模空间，重点是它们的同调和上同调。也就是说，它将考虑其点是d维流形的空间(因此，不是局部看起来像2维空间的表面，而是局部看起来像d维空间的空间)，并且在该空间中的运动对应于弯曲和拉伸。流形是几何学中研究的基本对象，因此，给定维度的所有流形的空间与这门学科中可以提出的许多问题密切相关。这个项目的一部分是使用和开发一个由Galatius和PI创建的强大的新工具，以便在一定的“稳定范围”内研究几何问题。此外，该项目将引入新的方法来理解这个“稳定范围”之外的流形的模空间，在那里缺乏系统的画面。

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.