Manifolds with Group Actions and their Quotients

摘要奖：DMS-0606869主要研究人员：丽贝卡·戈尔丁·奥比诺兹是最简单的奇异空间之一。它们在辛几何中作为约简出现，在代数几何中(在这里它们被称为Deligne-Mumford堆栈)作为某些模空间出现，在拓扑学中作为商空间出现。这些空间的代数不变量，如弦Betti数、扭曲的Hodgenbers数和Chen-Ruan上同调(也称为Orbiold上同调)最近引起了人们的兴趣，因为它们足够精细，可以看到奇点，在某些情况下描述了奇点的(Crepant)分解的上同调。本文将根据PI和合著者的工作探索新的方法来计算一个奥比福尔德的Chen-Ruan上同调，即奥比福尔德是一个紧的交换Lie群的整体商。这些方法将给出超曲簇的Chen-Ruan上同调(四元数n-空间的超kahler压缩)的新的组合公式，整体阿贝尔商的K-理论公式，以及非阿贝尔(李群)商的Chen-Ruan上同调。我们希望用这些方法来打折或证明关于或二叉形和折线解的猜想。在另一项工作中，PI正在为具有圆作用的流形和孤立不动点的等变上同调问题寻求一个通用的组合公式。我们找到了将某些正则类限制到任何其他不动点的拓扑公式，从而得到了一种六维分类。在高维，我们在流形是Kahler且具有不变的Palais-Smer度量(或具有其他相当刚性的结构，如GKM空间)的情况下，找到了一个“正”的限制公式。这项工作应该导致等变上同调的乘积结构的组合公式，这是一个广泛感兴趣的话题，因为它适用于旗形流形和环状变种，以及其他变种。在第三个项目中，我们建议使用在Bott-Samelson流形上发展的方法来研究Schubert演算中的结构常数的不同公式。这也应该推广到作用在G/B上的对称群(G上的对合的不动点)情况下的不动点限制公式。这项研究源于经典力学。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。对称性产生于这样一个事实，即物理对所有的观察者都是相同的。通过用对称性“折叠”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。在广义上，这项授权涉及用群体行动描述这些空间，以及它们的商。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由于对称性是如何“刚性”的。

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.