Heidelberg Symplektisches Seminar

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Summer semester 2019



Wed July 24 ∙  11:15  ∙  Sergei Tabachnikov (PennState)    Cross-ratio dynamics on ideal polygons

Define a relation between labeled ideal polygons in the hyperbolic 3-space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal a constant c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including a continuous version of this relation.



Wed July 17 ∙  16:00 In SRA  ∙  Alvaro del Pino (Utrecht)    Submanifolds of jet spaces and wrinkling

Wrinkling (and, relatedly, surgery of singularities) is one of the main tools in the theory of h-principles. In Contact Topology, wrinkling is best known for its role in the study of legendrians: E. Murphy's h-principle for loose legendrians and D. Álvarez-Gavela's simplification of front singularities both rely on it. Just like the space of 1-jets of functions is endowed with a canonical contact structure, higher jet spaces are endowed with other canonical distributions. I will report on joint work with L. Toussaint in which we use wrinkling to construct submanifolds of jet spaces tangent to these canonical distributions (generalising thus the study of legendrians). I will pay particular attention to explaining how the contact case differs from the general setup.



Wed July 17 ∙  11:15  ∙  Sergei Tabachnikov (PennState)    Frieze Patterns

Frieze patterns are interesting combinatorial objects introduced by Coxeter. Recently they have attracted much attention due to their relation with the theory of cluster algebras. I shall introduce frieze patterns and prove the theorem of Conway and Coxeter that relates arithmetical frieze patterns with triangulations of polygons. There is an intimate, and somewhat unexpected, relation between three object: frieze patterns, 2nd order linear difference equations, and polygons in the projective line. I shall describe some recent work on frieze patterns, including an interpretation of frieze patterns as a discretization of a coadjoint orbit of the Virasoro algebra.



Fri July 12 ∙  15:30  ∙  Room INF205  ∙  Christian Seidel (Bochum)   Closed geodesics in compact Lorentzian manifolds of splitting type

Let M=M0×S1 be a compact Lorentzian manifold of splitting type, equipped with a splitting metric. Provided that the fundamental group of M0 is finite, a result by Candela, Giannoni and Masiello ensures the existence of infinitely many closed timelike geodesics. I will present a quick view on their proof and discuss the case of an infinite fundamental group using some results by Galloway.



Wed July 10 ∙  11:15  ∙  Sergei Tabachnikov (PennState)    Elementary geometry is dead. Long live elementary geometry! (Part II)

Cf. Part I



Wed July 3 ∙  11:15  ∙  Sergei Tabachnikov (PennState)    Elementary geometry is dead. Long live elementary geometry! (Part I)

By “elementary”, I do not mean Euclidean axiomatic high school geometry, nor do I mean that the results that I will discuss are expected or easy to obtain. I use this term to distinguish my topic from differential geometry. I shall present a sampler of recent results that, in most cases, were discovered as a result in computer experiments and were motivated by the theory of completely integrable systems. The topics include the circumcenter of mass of polygons, the locus of centroids of Poncelet polygons, billiards in ellipses and Ivory’s lemma, Poncelet grid, a theorem of Kasner and its generalizations, projective configuration theorems, and a variation on Steiner’s porism. I shall also describe four equivalent properties of completely integrable billiards.



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

TBA.



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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