Mathematical Sciences: Topology and Manifolds

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9401443 McDuff Dusa McDuff: The study of symplectic topology has made great progress in the past decade, and is now sufficiently developed to have significant impact in other fields, for example the use of Floer homology in studying 3- and 4-dimensional manifolds, and, more recently, the application of J-holomorphic curves in the study of mirror symmetry and quantum cohomology. In recent years, McDuff has devoted her energies to developing the tools of symplectic topology with the main aim of understanding the topological structure of compact symplectic manifolds. She intends to continue working in the same general direction in the next three years, in an attempt to solve some of the very interesting problems which still remain. Lowell Jones: Jones will continue his collaborative efforts with F. T. Farrell, investigating rigidity properties of the classical closed spherical manifolds. In prior work they have shown that closed nonpositively curved manifolds and closed double coset manifolds are topologically rigid, in dimensions 5. In work in progress they show that any closed complete affine flat manifold is topologically rigid, in dimensions 5. They have also constructed many examples of strictly negatively curved closed manifolds which are homeomorphic to one another but not diffeomorphic, thus verifying that the class of nonpositively curved manifolds does not enjoy smooth rigidity. Whether or not the class of closed double coset manifolds, or the class of closed complete affine flat manifolds, are smoothly rigid remains an open question which Jones intends to pursue. Dusa McDuff: A symplectic structure is a very basic kind of geometric structure which underlies all the equations of classical and quantum physics. In the last ten years, new tools have been developed which allow mathematicians for the first time to gain some understanding of the global meaning of this kind of structure. One focus of McDuff's work is the geometric explora tion of the concept of energy. In physics this is the driving force of movement (of systems ranging from the planetary system to plasmas). The new methods of symplectic topology make it possible in some cases to estimate the minimum energy required to accomplish a certain movement. This work is very basic and should eventually have important implications in the study of physical systems. Another focus of her work is the attempt to understand what a symplectic structure really is by investigating the geometric properties of the spaces which are symplectic. The idea is that these spaces are the embodiment of the abstract idea of a symplectic structure. Lowell Jones: In 1829 and 1832 the mathematicians N. Lobachevsky and J. Bolyai independently solved a two thousand year old mathematical problem by verifying that Euclid's fifth axiom for plane geometry is independent from the first four of Euclid's axioms. They did this by investigating a new type of geometry having "negative curvature." Today the 2-dimensional geometries of Lobachevsky and Bolyai are important special cases of the larger class of geometric structures having "non-positive curvature," which can occur in all the higher dimensions, and which have been the focus of a tremendous amount of research over the last forty years. An important theme in this research has been that each such geometric structure has an associated algebraic structure (called the "fundamental group") which acts as a sort of genetic code for the underlying geometric structure. Although it is far from true that the fundamental group always determines the underlying geometric structure, this is true in some special cases. Jones has recently verified that the fundamental group always does completely determine the "topological nature" of the underlying geometric structure. To paraphrase this, Jones has shown that the fundamental group determines the rough outline of the underlying geometric structure, even though it does not de termine all the details of the geometric structure. ***