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Adelic models, rigidity and equivariant cohomology

Adelic 模型、刚性和等变上同调

基本信息

批准号：
EP/P031080/1
负责人：
John Greenlees
金额：
$ 40.62万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP031080%2F1
关键词：
Adelic models rigidity equivariant cohomology

项目摘要

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.

代数拓扑学旨在捕捉刚性代数不变量中几何问题的本质。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，那么能够查看所有的不变量，并选择那些结构特别好的不变量，或者那些捕捉到你关心的现象的不变量，是很有用的。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，所有的几何物体可能都有一个指定的旋转对称性，这是我们想要捕获的结构的一部分）。这个不变量集合具有所谓的张量三角化范畴（tt-category）的结构。该项目的目的是提供一种为tt类别构建模型的标准方法。这是一个两步的过程，首先确定一个对象，比如代数变量和它的函数环，然后确定额外的结构。一旦建立了标准模型，我们就给出标准来表明所得到的模型不仅是一个很好的近似，而且实际上恢复了整个tt类别（刚性）。当涉及到理性等变上同理论时，由于PI和合作者先前的工作，这一结果在环面和秩1群中是已知的。该方法的有效性的一个检验是，它应该适用于对广泛的其他紧李群给出上同调群的分析。在此过程中，该项目将研究tt范畴（幂上同调群）的新不变量，在熟悉的情况下计算它们，并了解它们在刚性中的作用。对tt类别采用统一方法的另一个好处是，来自一个领域的例子有时可以转移到另一个领域，我们的目标是考虑这种类型的几个例子。