Poisson Geometry and Riemannian Geometry

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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.