Celebratio Mathematica — McDuff

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Works connected to Leonid V. Polterovich

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D. McDuff and L. Polterovich: “Symplectic packings and algebraic geometry,” Invent. Math. 115 : 1 (1994), pp. 405–429. With an appendix by Yael Karshon. MR 1262938 Zbl 0833.53028 article

Leonid V. Polterovich	Related
Yael Karshon	Related

F. Lalonde, D. McDuff, and L. Polterovich: “On the flux conjectures,” pp. 69–85 in Geometry, topology, and dynamics (Montreal, 26–30 June 1995). Edited by F. Lalonde. CRM Proceedings Lecture Notes 15. American Mathematical Society (Providence, RI), 1998. MR 1619124 Zbl 0974.53062 ArXiv dg-ga/9706015 incollection

François Lalonde	Related
Leonid V. Polterovich	Related

F. Lalonde, D. McDuff, and L. Polterovich: “Topological rigidity of Hamiltonian loops and quantum homology,” Invent. Math. 135 : 2 (1999), pp. 369–385. MR 1666763 Zbl 0907.58004 article

This paper studies the question of when a loop \( \phi = \{\phi_t\}_{0\leq t\leq 1} \) in the group \( \mathrm{Symp}(M,\omega) \) of symplectomorphisms of a symplectic manifold \( (M,\omega) \) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if \( \phi \) is Hamiltonian with respect to \( \omega \), and if \( \phi^{\prime} \) is a small perturbation of \( \phi \) that preserves another symplectic form \( \omega^{\prime} \), then \( \phi^{\prime} \) is Hamiltonian with respect to \( \omega^{\prime} \). This allows us to get some new information on the structure of the flux group, i.e. the image of \( \pi_1(\mathrm{Symp}(M,\omega)) \) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of \( M \).

François Lalonde	Related
Leonid V. Polterovich	Related

L. Polterovich: “Focus on the scientist: Dusa McDuff,” MSRI Emissary (Spring 2010), pp. 3. article