Equivariant aspects in cohomology, K-theory and index theory

上同调、K 理论和指数理论中的等变方面

• 批准号: RGPIN-2014-06520
• 负责人: Franz, Matthias
• 金额: $ 1.02万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635425
• 关键词: Equivariant; aspects; cohomology; theory; index

The aim of the project is to study symmetries and their mathematical incarnations, group actions. We are mostly interested in manifolds with rotational symmetries, which correspond to actions of tori. Specifically, we will look at the effect of the symmetry on algebraic invariants of these spaces, notably on equivariant cohomology and equivariant K-theory.A major theme is centred around syzygies. The notion of syzygy from commutative algebra interpolates between torsion-free and free modules. In a joint project with Chris Allday (Hawaii) and Volker Puppe (Konstanz) I have initiated the study of syzygies in the context of torus-equivariant cohomology. Our approach unifies and generalizes many results concerning the freeness or torsion-freeness of equivariant cohomology, and it provides a better understanding of the equivariant Poincaré pairing and the so-called "GKM method" for computing equivariant cohomology.One objective of the present project is to extend this theory to other groups (including p-tori and non-commutative groups) and also to equivariant K-theory. Part of this will be carried out with Volker Puppe and Reyer Sjamaar (Cornell). I will also investigate a new class of torus manifolds which are emerging from the previous work and generalize the "polygon spaces" studied by many authors. These spaces are intersections of real quadrics, which provides connections to singularity theory, complex analysis and non-commutative geometry.Many spaces are what is called "equivariantly formal"; they often come up in combinatorics and representation theory. Here I want to find a common proof of equivariant formality for the two main classes of examples.Another objective is to explore the connections between our work on syzygies and recent work of De Concini, Procesi and Vergne about vector partition functions, splines and the equivariant index. Both are based on the same work of Atiyah and Singer. While there are similar patterns, the precise connection between the two is not understood so far.A smaller sub-project focuses on a topological approach to the Galois theory of Pythagorean fields via actions of 2-tori.

该项目的目的是研究对称性及其数学化身，小组行动。我们对旋转对称性的流形非常感兴趣，这与托里的动作相对应。具体来说，我们将研究对称性对这些空间代数不变的影响，特别是对等效的同类综合学和等效的K理论。一个主要主题以syzygies为中心。从无扭转和游离模块之间的交换代数中的Syzygy概念。在与Chris Allday（Hawaii）和Volker Puppe（Konstanz）的联合项目中，我在Torus-Equivariant共同体学的背景下启动了Syzygies的研究。我们的方法统一并概括了许多结果，这些结果涉及等效共同体的烦恼或扭转，它可以更好地理解对等效的Poincaré配对以及用于计算Equivariant同胞的所谓“ GKM方法”。其中一部分将与Volker Puppe和Reyer Sjamaar（Cornell）一起进行。我还将研究一类新的圆环歧管，这些歧管正在从先前的工作中浮现，并概括了许多作者研究的“多边形空间”。这些空间是真实四边形的交叉点，它们与奇异理论，复杂分析和非共同的几何形状提供了联系。许多空间就是所谓的“ equivariant operformally正式”。他们经常以组合和代表理论出现。在这里，我想为两个主要类别的示例找到一个常见的均等形式证明。另一个目标是探索我们在Syzygies上的工作与De Concini，Procesi和Vergne最近有关矢量分区功能，光谱和等效索引的最新工作之间的联系。两者都是基于Atiyah和Singer的同样作品。虽然存在类似的模式，但到目前为止，两者之间的精确连接尚不清楚。较小的子项目侧重于通过2-TORI的作用对毕达哥拉斯领域的Galois理论进行拓扑方法。