Symplectic Topology and Hamiltonian Dynamics

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Symplectic geometry is an intriguing generalization of Kahler geometry to a wide, and as yet not completely understood, class of smooth manifolds. Though it retains echoes of many of the structural features of the Kahler world (for example Lefschetz pencils, and complex curves),symplectic geometry is much more flexible than Kahler geometry.In particular, every finite dimensional symplectic manifold has an infinite dimensional group of structure-preserving transformations (called symplectomorphisms), while the corresponding group in the Kahler case is necessarily finite dimensional. McDuff proposes to study the topological properties of the whole symplectomorphism group, and in particular its relation to its finite dimensional subgroups. One important question is to develop a criterion for detecting if a given circle in the symplectomorphism group is homotopially trivial. McDuff has been working with Sue Tolman on this question and proposes to continue this collaboration, making a special study of the case of symplectic toric manifolds. In another separate project, she hopes to develop a fuller theory of symplectic characteristic classes to provide homological tools for understanding these questions. Both symplectic and Kahler geometry are very important in modern theoretical physics; string theories often study fields defined over a special kind of six dimensional Kahler space called a Calabi--Yau manifold, while instanton corrections to many equations and functions are often defined in purely symplectic terms. In order to understand a geometry it is essential to understand what kind of transformations preserve it; for example in the standard Euclidean geometry studied in high school the structure (distances and angle measurements) is preserved by rotations and translations. This project studies the properties of high dimensional families of these transformations (for example, the set of all rotations of the plane) rather than of individual transformations. There is a well understood theory (the theory of Lie groups) that works for rigid geometries such as Euclidean or Kahler geometry. One main aim of this project is to see how much of the structure remains in the more flabby and flexible symplectic world, where are infinitely many intrinsically different ways of perturbing space.