The Geometry and Dynamics of Symplectic Manifolds

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AbstractAward: DMS-0905191Principal Investigator: Dusa McDuffSince a symplectic structure allows one to measure the areas oftwo dimensional surfaces, it is natural that the two dimensionalsubmanifolds of a symplectic manifold are key elements of theirglobal structure. The choice of an auxiliary almost complexstructure J determines a specially interesting class of suchsurfaces, namely those that are J-holomorphic. Using them it ispossible to build many interesting homology theories, such asquantum cohomology or symplectic field theory. Recently veryintriguing connections have come to light concerning the relationbetween the dynamical properties of the symplectomorphisms on aspace (in particular, how much symmetry the space has) and thestructure of its quantum homology ring. McDuff recentlydiscovered that if the space has a circle symmetry then it isuniruled, which implies that for every choice of J there is aJ-holomorphic sphere though every point in the space. One of herproposed projects will investigate such connections in moredepth. Another will investigate the structure of toricmanifolds, which are symplectic manifolds with maximal abeliansymmetric group. She also proposes a joint project with Schlenkthat will illuminate a very basic symplectic rigidity phenomenon.This attempts to understand exactly when a four dimensionalsymplectic ellipsoid can be squeezed inside a ball. This givesrise to some very interesting number-theoretic questions, andalso indicates a connection between the combinatorics ofJ-holomorphic curves in the blow up of the projective plane andthe numbers that appear as indices in embedded contact homology.A space can have one of several fundamental geometric structures,for example a way of measuring distance and angle (as inEuclidean geometry) or a way of measuring the area of twodimensional objects (as in symplectic geometry.) The structuresstudied in symplectic geometry are important because they notonly underlie the equations of classical energy-conservingsystems such as the planetary system, but also appear as a vitalcomponent of many of the modern theories in physics such asstring theory. This project aims to further our basicunderstanding of symplectic spaces. One line of inquiryconcentrates on questions about the influence of structures inthe large (such as cohomology) on the dynamical properties of thespace, investigating for example the number and nature of thepoints that are fixed under an arbitrary movement of the space.Another line of inquiry investigates what one might think of asthe crystalline nature of small pieces of a symplectic space;under pressure how do such small pieces fold so as to take upless space? This second line of inquiry leads to some veryinteresting questions in elementary number theory andcombinatorics, the first appearance in symplectic geometry of arelation between these fields. These questions can be explainedto high school students, and so will provide an excellent way toexplain to young people something of what research mathematiciansdo today and to stimulate their interest in the field.