Equivariant aspects in cohomology, K-theory and index theory

该项目的目的是研究对称性及其数学化身，即群体行为。我们最感兴趣的是具有旋转对称性的流形，它对应于环面的作用。具体地说，我们将研究对称性对这些空间的代数不变量的影响，特别是对等变上同调和等变K-理论的影响。交换代数中的合子概念内插在无挠和自由模之间。在与Chris Allday(夏威夷)和Volker Puppe(Konstanz)的一个联合项目中，我启动了环面等变上同调背景下的合子研究。我们的方法统一和推广了许多关于等变上同调的自由性或无挠性的结果，并提供了对等变Poincaré配对和计算等变上同调的所谓“GKM方法”的更好的理解。**本项目的一个目的是将这一理论推广到其他群(包括p-环面群和非交换群)以及等变K-理论。这其中的一部分将与Volker Puppe和Reyer Sjamaar(康奈尔大学)一起进行。我还将研究从前人的工作中出现的一类新的环面流形，并推广许多作者所研究的“多边形空间”。这些空间是实二次曲面的交集，它提供了奇点理论、复分析和非交换几何的联系。**许多空间是所谓的“等变形式”；它们经常出现在组合学和表示论中。在这里，我想为这两类主要的例子找到一个等变形式化的通用证明。**另一个目标是探索我们在合子上的工作与de-Concini，Procesi和Vergne最近关于向量配分函数、样条和等变指数的工作之间的联系。两者都是基于阿提亚和辛格的相同工作。虽然有相似的模式，但两者之间的确切联系到目前为止还不清楚。**一个较小的子项目专注于通过2-Tori的作用对毕达哥拉斯场的伽罗瓦理论进行拓扑处理。

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.