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)
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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