Manifolds with Group Actions and their Quotients

摘要：dms -0606869项目负责人：Rebecca GoldinOrbifolds是最简单的奇异空间类型之一。它们在辛几何中作为约简出现，在代数几何中（它们被称为delign - mumford堆栈）作为某些模空间出现，在拓扑学中作为商空间出现。这些空间中的代数不变量，如弦贝蒂数、扭曲霍奇数和陈-阮上同调（也称为轨道上同调）最近引起了人们的兴趣，因为它们足够微妙，可以看到奇点，并且在某些情况下描述了奇点（creant）分辨率的上同调。本基金将探索计算轨道折叠陈-阮上同调的新方法，这是继PI和合作者在轨道折叠是紧的阿别列群的全局商的情况下所做的工作之后。这些方法将会得到一个关于高曲变元的陈-阮上同调的新组合公式，一个关于全局阿贝尔商的k理论公式，以及非阿贝尔（李群）商的陈-阮上同调的新组合公式。我们希望用这些方法来反驳或证明某些关于双折和皱纹决议的猜想。在另一项工作中，PI正在寻求一个关于具有圆作用和孤立不动点的流形的等变上同调问题的一般组合公式。我们找到了一类正则类对任意不动点的约束的拓扑公式，这就产生了一种六维的分类。在高维中，我们发现了一个“正”的限制公式，当流形是Kahler并且携带一个不变的palais - small度量（或者具有其他相当刚性的结构，例如GKM空间）。这项工作将导致在等变上同调中的乘积结构的组合公式，这是一个广泛感兴趣的话题，因为它适用于标志流形和环形品种，以及其他品种。在第三个项目中，我们建议使用基于bot - samelson流形演算的方法来研究舒伯特演算中结构常数的不同公式。这也可以推广到作用于G/B的对称群（G上的对合不动点）的不动点限制公式。这项研究植根于经典力学。相空间，由粒子的位置和动量组成，是辛空间的一个例子，具有对称群的作用，也称为哈密顿作用。对称源于这样一个事实，即物理学对所有观察者来说都是一样的。通过对称“坍缩”一个高维辛空间，我们可以得到一个更小的空间，有时更容易处理。从广义上讲，这个项目关注的是用群体行为及其商数来描述这些空间。它将对弦理论中出现的某些物理问题产生影响，并有助于讨论我们的物理宇宙是如何“刚性”的对称性。

AbstractAward: DMS-0606869Principal Investigator: Rebecca GoldinOrbifolds are among the simplest type of singular spaces. Theyarise in symplectic geometry as reductions, in algebraic geometry(where they are called Deligne-Mumford stacks) as certain modulispaces, and in topology as quotient spaces. Algebraic invariantsof these spaces, such as stringy Betti numbers, twisted Hodgenumbers, and Chen-Ruan cohomology (also called orbifoldcohomology) have recently gained interest because they are subtleenough to see the singularities, and in some cases describe thecohomology of a (crepant) resolution of singularities. This grantwill explore new methods to compute the Chen-Ruan cohomology ofan orbifold, following work done by the PI and coauthors in thecase that the orbifold is a global quotient by a compact abelianLie group. These methods should produce a new combinatorialfomula for the Chen-Ruan cohomology of a hypertoric variety (ahyperkahler reduction of quaternionic n-space by a compacttorus), a formula for the K-theory of global abelian quotients,and the Chen-Ruan cohomology of non-abelian (Lie group)quotients. We hope to use these methods to discount or provecertain conjectures about or bifolds and crepant resolutions. Ina separate undertaking, the PI is pursuing a generalcombinatorial formula for questions about the equivariantcohomology of manifolds with circle actions and isolated fixedpoints. We have found a topological formula for the restrictionof certain canonical classes to any other fixed point, which hasled to a kind of classification in 6-dimensions. In higherdimensions, we have found a "positive" restriction formula in thecase that the manifold is Kahler and carries an invariantPalais-Smale metric (or has other rather rigid structure, such asbeing a GKM space). This work should lead to combinatorialformulas for the product structure in the equivariant cohomology,a topic of broad interest because it applies to flag manifoldsand toric varieties, among other varieties. In a third project,we propose to investigate different formulae for the structureconstants in Schubert calculus using methods developed on thecalculus of Bott-Samelson manifolds. This should also generalizeto fixed-point restriction formulas in the case of a symmetricgroup (the fixed points of an involution on G) acting on G/B.This research has roots in classical mechanics. Phase space,which consists of the position and momentum of a particle, is anexample of a symplectic space, with an action by the symmetrygroup, also called a Hamiltonian action. Symmetry arises from thefact that the physics is the same for all observers. By"collapsing" a high-dimensional symplectic space by thesymmetry, we can get a smaller space that is sometimes easier towork with. In a broad sense, this grant is concerned withdescribing these spaces with group actions, and theirquotients. It will have impact on certain questions in physicsthat have arisen in string theory, as well as contribute to thediscussion of how "rigid" our physical universe is byvirtue of symmetry.