Mathematical Sciences: Topology and Manifolds

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Professor McDuff has recently concentrated her attention on 4-dimensional manifolds, using and developing techniques first introduced by Gromov. She has shown that a large class of minimal symplectic 4-manifolds (those containing "nice" symplectically embedded 2-spheres) carry a unique symplectic structure in each cohomology class. She intends to continue this work: for example, to see if she can extend the above result to manifolds which contain a symplectically embedded surface of higher genus, or to see if she can find a 4-manifold which supports different symplectic and Kahler structures in the same cohomology class. The class of symplectic 4-manifolds lies between the class of Kahler surfaces and the class of smooth 4- manifolds, and, if it turns out that symplectic 4-manifolds behave nicely enough, one might be able to use information on their structure to obtain information on the class of 4-manifolds itself. Professor Jones will continue his joint research with F. T. Farrell. One of their most recent results states that the surgery L-groups and the rational algebraic K-groups for the group ring ZG can be computed in a simple way from the L-groups and rational K-groups of group rings ZG', where G' is any virtually cyclic subgroup G' in G provided G is a co-compact discrete subgroup of a virtually connected Lie group. In the next three years they hope to prove this same result for more general groups G. Symplectic manifolds arise in the study of mechanics, both classical and quantum. Lie groups are also ubiquitous in mechanics. It would seem that mathematics and theoretical physics are not entirely distinct subjects at this level.