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Lectures on groups of symplectomorphisms. (English) Zbl 1078.53088

Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 43-78 (2004).
Summary: These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srní, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds \((M,\omega)\) 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 \(M=\mathbb{C} \mathbb{P}^2\) and \(M=\mathbb{S}^2\times S^2\). 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 \(\text{Ham}(M)\). 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 \(\text{Ham}(M,\omega)\) that minimize the Hofer norm over all paths with the given endpoints.
For the entire collection see [Zbl 1034.53002].

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension