Heidelberg Symplektisches Seminar

Summer semester 2019

Wed May 22  ∙  11:00  ∙  Markus Reineke (Bochum)    Cohomological Hall Algebras


Wed May 29  ∙  11:00  ∙  Pazit Haim-Kislev (Tel Aviv)    Closed characteristics on the boundary of convex polytopes

We introduce a simplification to the problem of finding a closed characteristic with minimal action on the boundary of a convex polytope in R2n, which yields a combinatorial formula for the EHZ capacity. As an application, we show that the EHZ capacity of a general convex body is sub-additive with respect to hyperplane cuts, and we bound the systolic ratio for simplices in R4.

Winter semester 2018/19

Wed Feb 6  ∙  Silvia Sabatini (Köln)    Hamiltonian S1-spaces with large minimal Chern number

Consider a compact symplectic manifold of dimension 2n which is acted on by a circle in a Hamiltonian way with isolated fixed points; we refer to it as a Hamiltonian S1-space. In [S] it is proved that the minimal Chern number N is bounded above by n+1, bound which is expected for all positive monotone compact symplectic manifolds. Assuming that the Hamiltonian S1-space is monotone (i.e. the first Chern class is a multiple of the class of the symplectic form) in [GHS] several bounds on the Betti numbers are proved, these bounds depending on N. I will first discuss the ideas behind the proofs of the aforementioned facts, and then concentrate on N=n+1. In this case my student Isabelle Charton [C] proved that the manifold must be homotopically equivalent to a complex projective space of dimension 2n.

[C] Charton, "Hamiltonian manifolds with high index". Master thesis. University of Cologne, 2017.
[GHS] Godinho, von Heymann, Sabatini "12, 24 and beyond ", Advances in Mathematics, 319 (2017), 472 - 521.
[S] Sabatini "On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action", Communications in Contemporary Mathematics, 19, No. 04 (2017).

Wed Dec 12  ∙  Matthias Meiwes (Tel Aviv)    Wrapped Floer homology and surgery

Adapting a construction of Viterbo, Abouzaid and Seidel defined a map V between the wrapped Floer homologies of two pairs (M,L)⊂(M',L') where M,M' are Liouville domains and L,L' suitable exact Lagrangians in M and M'. V is an isomorphims if M' is obtained by attaching a subcritical handle on M or a handle on a Legendrian that is loose in the complement of the boundary of L in M. I will talk about the construction of V and some consequences.

Wed Nov 21  ∙  Kevin Wiegand (Gießen)    Odd-symplectic surgery

As discovered by Hofer the moduli space of holomorphic discs is related to the question of periodic Reeb orbits. A surgery construction leads to a cobordism theory with amazing properties. Holomorphic discs in the upper boundary yield to statements about non-dense characteristics in the lower boundary.

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