Poisson Geometry and Riemannian Geometry

Alan Weinstein-项目摘要：泊松几何与量化主要研究者：泊松结构是可微流形上的“量子非对易性的阴影”，表现为沿着叶具有辛结构的奇异叶理。 温斯坦将研究泊松几何的几个方面及其与非交换代数的关系。 该项目的中心部分，出于辛几何和泊松几何中凸性定理的研究，实际上属于黎曼几何。 Ulam关于群的近似同态和紧群作用的近似不变子流形的“广义稳定性问题”的版本已被归结为子流形族的平均问题。 然后将平均方法应用于真李群胚的分类，并应用于凸性理论和等参子流形几何。 该项目的一个独立部分将使用狄拉克结构的几何学和量子化（沿着叶子沿着具有前辛结构的奇异叶子），完成量子环面的分类直到森田等价。 最初，人们认为物理或数学对象的对称性是一种变换，它使对象的结构保持不变。 给定对象的对称性的集合，加上一个接一个地应用对称性的操作，形成了一个“对称群”，对它的分析有助于对所研究对象的研究。 许多重要的应用需要对称群概念的推广，甚至对称性本身的概念。 首先，人们只在对象的一部分上定义了对称性（例如，在复杂化合物中的单个分子之间），或者只使对象的属性近似不变的对称性（例如，在真实数据的研究中，其对称性通常是不完美的）。 通过平均化的方法，我们打算从不完美的对称性中推导出真正的对称性;这种理想化可以揭示物体的潜在结构，尽管我们对它的感知并不完美。平均化的方法也可以应用于计算机图形学中的问题，例如通过连续的曲线或曲面族将一条曲线或曲面“变形”为另一条曲线或曲面。 该项目的一部分涉及代数系统有关的那些通常的结合法律组成：（ab）c = a（bc）是满足只有近似。 这种系统最近被证明与假设的“M理论”有关，这可能会揭开高能物理的一些奥秘。

Alan Weinstein----------------------------------------------------------------------ABSTRACT OF PROJECT: POISSON GEOMETRY AND QUANTIZATIONPrincipal Investigator: Alan WeinsteinPoisson structures are the ``shadow of quantum noncommutativity'' ondifferentiable manifolds, manifested as singular foliations withsymplectic structures along the leaves. Weinstein will investigateseveral aspects of Poisson geometry and its relations tononcommutative algebra. The central part of the project, motivated bythe study of convexity theorems in symplectic and Poisson geometry,actually belongs to riemannian geometry. Versions of Ulam's ``generalstability problem'' concerning approximate homomorphisms from groupsand approximately invariant submanifolds for compact group actionshave been reduced to averaging problems for families of submanifolds.Several methods of averaging will be investigated. The averagingmethods will then be applied to the classification of proper Liegroupoids, with applications to convexity theory and isoparametricsubmanifold geometry. A separate part of the project will use thegeometry and quantization of Dirac structures (singular foliationswith presymplectic structures along the leaves), to complete theclassification of quantum tori up to Morita equivalence.This project is concerned with various extensions of the notions ofsymmetry and symmetry group. Initially, one thinks of a symmetry of aphysical or mathematical object as a transformation which leaves thestructure of the object invariant. The collection of symmetries of agiven object, together with the operation of applying one symmetryafter another, forms a "symmetry group" whose analysis aids the studyof the object under investigation. Many important applicationsrequire generalizations of the notion of symmetry group, and even thenotion of symmetry itself. First of all, one has symmetries definedonly on part of an object (e.g. between individual molecules in acomplex compound, rather) or symmetries which leave the properties ofan object only approximately invariant (e.g. in the study of realdata, whose symmetries are generally imperfect). Through a method ofaveraging, we intend to derive true symmetries from imperfect ones;this idealization could reveal the underlying structure of an objectdespite our imperfect perception of it. The method of averaging mayalso be applicable to problems in computer graphics, such as the"morphing" of one curve or surface into another by a continuous familyof curves or surfaces. Part of the project involves algebraic systemsrelated to those in which the usual associative law of composition:(ab)c = a(bc) is satisfied only approximately. Such systems haverecently been shown relevant to the hypothetical "M-theory" which mayunravel some of the mysteries of high-energy physics.