Lectures on Groups of Symplectomorphisms

Details Lectures on Groups of Symplectomorphisms Lectures on Groups of Symplectomorphisms

2002-01-06

arxiv.org/abs/math/0201032v2

Rendiconti del Circolo Matematico di Palermo, Ser II, Suppl 72(2004), 43--78

Mathematics

Symplectic Geometry

significantly revised, 36 pages; notes from summer school in Paris 2001 and winter school in Srni 2003

Scientific paper

These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srni, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om) from various points of view. Lectures 1 and 2 provide an overview of our current knowledge of their algebraic, geometric and homotopy theoretic properties. Lecture 3 sketches the arguments used by Gromov, Abreu and Abreu-McDuff to figure out the rational homotopy type of these groups in the cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve techniques. Much of the recent progress in understanding the geometry and topology of these groups has come from studying the properties of fibrations with the manifold M as fiber and structural group equal either to the symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base is S^2 has proved particularly important. Lecture 4 describes the geometry of Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten invariants via the Seidel representation. It ends by sketching Entov's explanation of the ABW inequalities for eigenvalues of products of special unitary matrices. Finally in Lecture 6 we apply the ideas developed in the previous two lectures to demonstrate the existence of (short) paths in Ham(M,\om) that minimize the Hofer norm over all paths with the given endpoints.

Affiliated with

Also associated with

No associations

Rating

Lectures on Groups of Symplectomorphisms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Lectures on Groups of Symplectomorphisms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lectures on Groups of Symplectomorphisms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-693497