Adelic models, rigidity and equivariant cohomology

代数拓扑学的目标是在严格的代数不变量中捕捉几何问题的本质。事实上，不变量本身就是一个合理的研究主题：如果人们试图找到好的不变量，那么能够查看所有不变量并选择那些结构特别好的不变量，或者捕获你关心的现象是有用的。在这个项目中，一个特别有趣的例子是当我们考虑对称性时（例如，所有的几何对象都可能具有指定的旋转对称性，这是我们想要捕获的结构的一部分）。这个不变量的集合具有所谓的张量三角范畴（tt-范畴）的结构。这个项目的目的是给出一个标准的方法来建立一个tt-范畴的模型。这是一个两步的过程，首先确定一个对象，如代数簇及其函数环，然后还确定额外的结构。一旦标准模型被构建，我们给出的标准表明，由此产生的模型不仅是一个很好的近似，但实际上恢复了整个TT-范畴（刚性）。当涉及到有理等变上同调理论时，由于PI和合作者以前的工作，这个结果对于环面和秩1群是已知的。这个项目的方法的有效性的测试是，它应该适用于给一个分析的上同调群的广泛的其他紧李群。沿着的方式，该项目将研究新的不变量的一个tt-范畴（adelic上同调群），计算他们在熟悉的情况下，并了解他们的作用，刚性。TT-范畴的统一方法的另一个好处是，来自一个竞技场的例子有时可以被转移到另一个竞技场，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.