Symplectic Topology

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

DMS-0072512 Dusa McDuff McDuff will continue studying the structure of symplectic manifolds and of the group G(M) of diffeomorphisms that preserve the symplectic structure on a given manifold M. Working together with Francois Lalonde, she recently discovered that these diffeomorphisms do not twist the manifold topologically very much. Indeed, using ideas from quantum homology, they have shown that the homotopy groups of the Hamiltonian subgroup H of G(M) act trivially on the rational homology of the underlying manifold M. The conjecture is that any fiber bundle with structural group H is a product as far as its rational cohomology is concerned. She also plans to study other topological implications of the existence of quantum homology, in particular its consequences for the Flux conjecture. A symplectic structure is a very basic structure on space that underlies the equations of classical physics. They have become very prominent recently because of their appearance in modern theories of duality, especially in the "mirror symmetry" phenomena of high energy physics. Mathematically, they are very interesting and help in understanding 4 dimensional spaces (curved space-times). Recently McDuff has focussed her attention on studying the different ways that the points of a space can move while still preserving this structure. Working with a collaborator, she has found that some ideas coming from string theory (known in the field as "quantum homology") show that spaces with symplectic structures are much more rigid than ones without, and cannot be moved and twisted up very much. She intends to pursue this line of questioning during the period of the grant. Many very interesting symplectic rigidity phenomena have been discovered, but there are still plenty of open questions.