The Geometry and Dynamics of Symplectic Manifolds

Barnard College, New York NY

Investigators

Abstract

Abstract Award: DMS-0905191 Principal Investigator: Dusa McDuff Since a symplectic structure allows one to measure the areas of two dimensional surfaces, it is natural that the two dimensional submanifolds of a symplectic manifold are key elements of their global structure. The choice of an auxiliary almost complex structure J determines a specially interesting class of such surfaces, namely those that are J-holomorphic. Using them it is possible to build many interesting homology theories, such as quantum cohomology or symplectic field theory. Recently very intriguing connections have come to light concerning the relation between the dynamical properties of the symplectomorphisms on a space (in particular, how much symmetry the space has) and the structure of its quantum homology ring. McDuff recently discovered that if the space has a circle symmetry then it is uniruled, which implies that for every choice of J there is a J-holomorphic sphere though every point in the space. One of her proposed projects will investigate such connections in more depth. Another will investigate the structure of toric manifolds, which are symplectic manifolds with maximal abelian symmetric group. She also proposes a joint project with Schlenk that will illuminate a very basic symplectic rigidity phenomenon. This attempts to understand exactly when a four dimensional symplectic ellipsoid can be squeezed inside a ball. This gives rise to some very interesting number-theoretic questions, and also indicates a connection between the combinatorics of J-holomorphic curves in the blow up of the projective plane and the 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 in Euclidean geometry) or a way of measuring the area of two dimensional objects (as in symplectic geometry.) The structures studied in symplectic geometry are important because they not only underlie the equations of classical energy-conserving systems such as the planetary system, but also appear as a vital component of many of the modern theories in physics such as string theory. This project aims to further our basic understanding of symplectic spaces. One line of inquiry concentrates on questions about the influence of structures in the large (such as cohomology) on the dynamical properties of the space, investigating for example the number and nature of the points that are fixed under an arbitrary movement of the space. Another line of inquiry investigates what one might think of as the crystalline nature of small pieces of a symplectic space; under pressure how do such small pieces fold so as to take up less space? This second line of inquiry leads to some very interesting questions in elementary number theory and combinatorics, the first appearance in symplectic geometry of a relation between these fields. These questions can be explained to high school students, and so will provide an excellent way to explain to young people something of what research mathematicians do today and to stimulate their interest in the field.