The Topology of Symplectomorphism Groups

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AbstractAward: DMS-0604769Principal Investigator: Dusa McDuff Symplectic manifolds all look alike at small scales, which meansthat they have many symplectomorphisms, that is, structurepreserving self-mappings. These form very interesting groups,which one can try to understand from a topological/algebraicviewpoint as well as a geometric viewpoint. McDuff has beenstudying their structure for several years, and proposes tocontinue this work. One project is to extend her analysis (inprevious joint work with Abreu) of the symplectomorphism groupfor rational/ruled manifolds of dimension four to similarmanifolds in higher dimension. Many new problems arise herebecause the manifold is no longer fibered by pseudoholomorphiccurves. She proposes to study the internal structure ofsymplectic manifolds (such as spaces of embedded symplectic ballsor Lagrangian tori), to see how this is reflected in thestructure of the group of symplectomorphisms. She also proposesa joint project with Hofer, which would develop a theory ofbranched manifolds that could be used in an equivariant versionof the new polyfold theory. In order to understand our physical world it is important todevelop a variety of ways of measuring objects in space. Themost familiar is the measurement of distances and angles, whichleads to the usual notions of geometry. This proposal concerns adifferent kind of geometry, called symplectic geometry, that isbased on measurements of two-dimensional objects. As firstdiscovered by Hamilton over 150 years ago, many spaces ofinterest to physicists have this kind of structure. For example,it underlies the equations of motion of energy-conserving systemssuch as the planetary system, since here each position coordinateis paired with a momentum coordinate to form a basictwo-dimensional object. Symplectic geometry also has muchrelevance to modern theories of physics such as string theory andmirror symmetry. In recent years there has been a very importantexchange of ideas between mathematics and physics that is largelyexpressed in symplectic terms. Hence it is very important tofurther our understanding of the fundamentals of symplecticgeometry. This project will develop our basic knowledge in thisarea, concentrating particularly on understanding the propertiesof the group of motions that preserve this geometry. It willexploit the new invariants coming from physics (such as quantumcohomology and symplectic field theory) in order to understandthe topological structure of these groups. Besides itstheoretical interest, the new symplectic geometry has led to thedevelopment of better methods for computing Hamiltonian motionssuch as the orbit of a satellite.