Bonn Symplectic Geometry Seminar

Abstract: I will start by explaining how, given a Morse function and a metric on a manifold M, one can get a Morse-theoretic construction of its fundamental group. I will then explain how to build "continuation maps" which give us functorial properties for the Morse fundamental group. Finally, I will adapt this discussion to the study of Legendrian submanifolds of contact manifolds and their Lagrangian fillings.

Abstract: Given a symplectic 4-manifold it may admit multiple toric fibrations. These can be seen as boundary points of the moduli space of almost toric fibrations. We will sketch that all toric fibrations are in the same connected component of this moduli space, as suggested by Symington. For this we use some results from tropical geometry.

Abstract: The fillability of Legendrian submanifolds is a classical problem in contact topology. Given a Legendrian submanifold of a contact manifold, we ask whether it admits an embedded exact Lagrangian filling. We generalise the construction of
Legendrian twist knots to arbitrary dimensions using the Legendrian clasp move and prove that every non-loose Legendrian submanifold gives rise to an infinite family of non-loose, non-fillable Legendrian submanifolds. This suggests that Legendrian submanifolds of contactisations rarely admit exact Lagrangian fillings in the corresponding symplectisations.

Our main method is to study how the clasp move affects the differential of the Chekanov–Eliashberg algebra. To this end, we establish a correspondence between moduli spaces of J-holomorphic curves with boundary and possibly multiple positive punctures that contribute to rational relative symplectic field theory (RRSFT). This leads to a wide-open question of the amount of information about a connected Legendrian submanifold that can be extracted from RRSFT. Especially, can there be any generalisations of finite-type invariants of knots to Legendrian submanifolds of higher-dimension?

Abstract:  In this talk, I will present a counterexample to the Viterbo conjecture. The conjecture predicts that Viterbo's spectral metric is bounded on the space of Lagrangians Hamiltonian isotopic to the zero section in the disk cotangent bundle D^*Q for all manifold Q. I will first go through the definition of Viterbo's spectral metric and then explain how a magnetic Hamiltonian can, under some topological assumption on Q, yield a counterexample to the conjecture. I will also sketch the proof of the complete classifications of the surfaces for which the Viterbo conjecture holds. This is joint work  with Filip Broćić and Ibrahim Trifa.

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