Adelic models, rigidity and equivariant cohomology

代数拓扑学的目的是用刚性代数不变量来刻画几何问题的本质。事实上，不变量本身是一个合理的研究主题：如果一个人试图找到好的不变量，能够查看所有不变量并选择那些结构特别良好的不变量，或者选择那些捕捉到您关心的现象的不变量，这是很有用的。在这个项目中特别感兴趣的一个例子是当我们考虑对称性时(例如，所有几何对象可能具有指定的旋转对称性，这是我们想要捕捉的结构的一部分)。这个不变量集合具有所谓的张量三角范畴(TT范畴)的结构。该项目的目的是给出一种为TT类别构建模型的标准方法。这是一个两步的过程，首先识别一个像代数变种一样的对象及其函数环，然后还识别额外的结构。一旦构建了标准模型，我们就给出了证明所得到的模型不仅是一个良好的近似，而且实际上恢复了整个TT范畴(刚性)的准则。当涉及到有理等变上同调理论时，这一结果对于Tori和RANK 1群是已知的，这要归功于PI和合作者之前的工作。这个项目方法的有效性的一个检验是，它应该应用于给出一系列其他紧李群的上同调群的分析。在此过程中，该项目将研究TT-范畴(adelic上同调群)的新不变量，计算常见情况下的不变量，并了解它们在刚性中的作用。对TT-范畴采取统一方法的另一个好处是，有时可以将一个领域的例子转移到另一个领域，我们的目标是考虑几个这种类型的例子。

Algebraic topology aims to capture the essence of geometric problems in rigid algebraic invariants. In fact the invariants themselves are a legitimate subject of study: if one seeks to find good invariants, it is useful to be able to look at all invariants and choose those that are particularly well structured, or which capture the phenomena you care about. An example of particular interest in this project is when we take symmetries into account (for example all of the geometric objects might have a specified rotational symmetry that is part of the structure we want to capture). This collection of invariants has the structure of a so-called tensor-triangulated category (tt-category). The aim of the project is to give a standard way of building a model for a tt-category. This is a two step process, first identifying an object like an algebraic variety and its ring of functions, but then also identifying additional structure. Once the standard model is constructed we give criteria for showing that the resulting model is not just a good approximation but actually recovers the whole tt-category (rigidity). When it comes to rational equivariant cohomology theories this result is known for tori and rank 1 groups thanks to previous work of the PI and collaborators. A test of the effectiveness of the methods of this project is that it should apply to give an analysis of cohomology groups for a wide range of other compact Lie groups. Along the way, the project will study new invariants of a tt-category (adelic cohomology groups), calculate them in familiar cases and understand their role in rigidity. Another benefit of the uniform approach to tt-categories is that examples from one arena can sometimes be transported to another, and we aim to consider several examples of this type.