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From Conservative Dynamics to Symplectic and Contact Topology
• Hansjörg Geiges (Universität zu Köln): contact geometry and low-dimensional topology
• Viktor Ginzburg (UC Santa Cruz): symplectic topology and Hamiltonian dynamical systems
• Federica Pasquotto (VU University Amsterdam): symplectic and contact topology
• Bob Rink (VU University Amsterdam): mechanics and conservative dynamics
• Rob Vandervorst (VU University Amsterdam): dynamical systems and variational methods
The aim of this workshop was to focus on some aspects of symplectic and contact geometry where
the interactions of these fields have proved most fruitful and have generated problems of high current interest. Exemplary for such questions are the Weinstein conjecture and the topology of the group of symplectic/Hamiltonian/contact diffeomorphisms.
The Weinstein conjecture is concerned with the existence of closed orbits of the Reeb vector field associated with a contact form, inspired by a result of Rabinowitz on periodic solutions of Hamiltonian systems. A wealth of techniques have been brought to bear on this conjecture since its original formulation in 1979, leading to various partial solutions. Hofer's approach via pseudoholomorphic curves was the first to yield significant results in dimension 3; this case has recently been resolved completely by Taubes, using Seiberg-Witten theory.
With the aid of topological techniques such as open book decompositions, the scope of Hofer's method has been extended significantly, also to higher dimensions. During the workshop,
several talks addressed the topic of the Weinstein conjecture. Abouzaid discussed Floer theoretic invariants associated to an exact Lagrangian submanifold of a Weinstein manifold and how this can be used to deduce existence of closed Reeb orbits (and thus verify the Weinstein conjecture) from existence of Reeb chords. Niederkrüger explained how Hofer's proof of the Weinstein conjecture for 3-manifolds with non-vanishing second homotopy group can be generalized to manifolds of dimension five admitting an embedded 3-sphere with a trivial Legendrian open book and which represents a non-trivial homotopy class (joint work with Massot and Wendl).
In his talk, Zehmisch addressed the existence of null-homologous Reeb links (known as the strong Weinstein conjecture) and discussed some results he obtained together with Geiges. In particular, they prove the conjecture for severel classes of manifolds (for instance, manifolds admitting a Giroux open book with subcritical pages).
The group of Hamiltonian/symplectic (resp. contact) diffeomorphism of a symplectic (resp. contact) manifold can be interpreted classically as the symmetries of a mechanical system. Two of the most
influential methods in the study of the group of Hamiltonian diffeomorphisms have been Hofer's invention of a metric on this group, and the recent construction of quasi-morphisms on this group by Polterovich-Entov, using Floer theory. During the workshop, Sandon explained the
construction of a bi-invariant metric, called the discriminant metric, on the universal cover of the contactomorphism group of any contact manifold (joint work with Colin). She also discussed the relation of this metric to other contact rigidity phenomena, such as non-squeezing. A closely related construction was presented by Albers, using the Rabinowitz action functional.
Further interesting interactions of symplectic/contact geometry with other fields of mathematics
were addressed in the talks of Ostrover (geometric analysis), van Koert (celestial mechanics),
Polterovich (quantum mechanics), and McLean (algebraic geometry).
The workshop lasted 5 days, with on average four 45 minutes lectures per day. There were two
"Short presentations" session, on Monday and Tuesday, during which graduate students were
given the opportunity to introduce themselves to the rest of the participants. The program contained
a lot of free time, which allowed for plenty of interaction and collaboration.
The following participants gave a lecture: Mohammed Abouzaid (New York), Peter Albers (Münster), Frèdèric Bourgeois (Bruxelles), Lev Buhovski (Chicago), Albert Fathi (Lyon), Basak Güurel (Nashville), Janko Latschev (Hamburg), Samuel Lisi (Bruxelles), Mark Mclean (Cambridge, US), Klaus Niederkrüger (Toulouse), Yaron Ostrover (Tel Aviv), Leonid Polterovich (Chicago), Sheila Sandon (Nantes), Andras Stipsicz (Budapest), Michael Usher (Athens, US), Otto Van Koert (Seoul), Kai Zehmisch (Köln, Germany).
The following junior participants delivered a short presentation:
Marta Batoreo (Santa Cruz), Matthew Strom Borman (Chicago), Max Dörner (Köln),
Doris Hein (Santa Cruz), Wyatt Howard (Santa Cruz), Arun Maiti (Leipzig), Thomas Rot (Amsterdam), Frol Zapolsky (München).
The following people also participated: Fan Ding (Beijing), Will J. Merry (Cambridge, UK), Egor Shelukhin (Tel Aviv), Oldrich Spacil (Aberdeen), Fabian Ziltener (Seoul).
Assessment and participants' comments
The quality of the talks during the workshop was extremely good and the topics managed to generate a lot of interest, as demonstrated by the fact that every talk was followed by very animated discussions. The speakers were very happy with the format of 45 minutes' talks. In fact, everyone appreciated the fact that the workshop program left ample time for interaction, which in turn created significant new opportunities for exchanges and collaboration. The paper A discontinuous capacity by Zehmisch and Ziltener (arXiv:1208.6000 [math.SG]) is a direct result of the workshop.
The facilities and support of the Lorentz Center and the accommodation provided were excellent.
A particularly successful component of the workshop program were the Short presentations delivered by graduate students.
The participants' comments confirm our impressions. One of the speakers wrote to us: "The academic atmosphere at the conference was very high. To get the possibility to work at the Lorenz Centre was one of the fantastic features...", while
another speaker wrote: "Among the participants were not only leading researchers in the field but also a good portion of junior researchers. The latter group presented their research in short talks which turned out to be of very fruitful format...
As a matter of fact the amount of interaction before, between and after talks was astonishing. This certainly was emphasized by the excellent research atmosphere of the Lorentz Center... As for my own research the conference was enormously valuable. My talk was a report on ongoing work and it is fair to say that it leaped ahead as a consequence of the conference..."
Apart from the facilities, secretarial support and financial contribution of the Lorentz Center, we obtained funding from the cluster NDNS+ (2500 Euro) and from the European network CAST (5000 Euro).
Hansjörg Geiges (Köln, Germany)
Viktor Ginzburg (Santa Cruz, United States)
Federica Pasquotto (Amsterdam, Netherlands)
Bob Rink (Amsterdam, Netherlands)
Rob Vandervorst (Amsterdam, Netherlands)