The Topology of Symplectomorphism Groups

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Abstract Award: DMS-0604769 Principal Investigator: Dusa McDuff Symplectic manifolds all look alike at small scales, which means that they have many symplectomorphisms, that is, structure preserving self-mappings. These form very interesting groups, which one can try to understand from a topological/algebraic viewpoint as well as a geometric viewpoint. McDuff has been studying their structure for several years, and proposes to continue this work. One project is to extend her analysis (in previous joint work with Abreu) of the symplectomorphism group for rational/ruled manifolds of dimension four to similar manifolds in higher dimension. Many new problems arise here because the manifold is no longer fibered by pseudoholomorphic curves. She proposes to study the internal structure of symplectic manifolds (such as spaces of embedded symplectic balls or Lagrangian tori), to see how this is reflected in the structure of the group of symplectomorphisms. She also proposes a joint project with Hofer, which would develop a theory of branched manifolds that could be used in an equivariant version of the new polyfold theory. In order to understand our physical world it is important to develop a variety of ways of measuring objects in space. The most familiar is the measurement of distances and angles, which leads to the usual notions of geometry. This proposal concerns a different kind of geometry, called symplectic geometry, that is based on measurements of two-dimensional objects. As first discovered by Hamilton over 150 years ago, many spaces of interest to physicists have this kind of structure. For example, it underlies the equations of motion of energy-conserving systems such as the planetary system, since here each position coordinate is paired with a momentum coordinate to form a basic two-dimensional object. Symplectic geometry also has much relevance to modern theories of physics such as string theory and mirror symmetry. In recent years there has been a very important exchange of ideas between mathematics and physics that is largely expressed in symplectic terms. Hence it is very important to further our understanding of the fundamentals of symplectic geometry. This project will develop our basic knowledge in this area, concentrating particularly on understanding the properties of the group of motions that preserve this geometry. It will exploit the new invariants coming from physics (such as quantum cohomology and symplectic field theory) in order to understand the topological structure of these groups. Besides its theoretical interest, the new symplectic geometry has led to the development of better methods for computing Hamiltonian motions such as the orbit of a satellite.