Advances in Symplectic topology (Virtual)

List of records

1. Umberto Hryniewicz (RWTH-Aachen): Reeb flows in dimension three with exactly two periodic orbits

17/05/2021 15:00

In this talk I will present a complete characterization of Reeb flows on closed 3-manifolds with precisely two periodic orbits. The main step consists in showing that a contact form with exactly two periodic Reeb orbits is non-degenerate. The proof combines the ECH volume formula with a study of the behavior of the ECH index under non-degenerate perturbations of the contact form. As a...

2. Jo Nelson (Rice University): Embedded Contact Homology of Prequantization Bundles

17/05/2021 16:30

In 2011, Farris provided a means of computing Z_2-graded embedded contact homology (ECH) of prequantization bundles over Riemann surfaces, producing an isomorphism between ECH of the bundle and the exterior algebra of the homology of the base. In joint work with Morgan Weiler, we upgrade to a full Z-grading on the chain complex and obtain a stabilization result. We additionally explain how...

5. Ana Rechtman (Université de Strasbourg): Broken books and Reeb dynamics in dimension 3

18/05/2021 15:00

Giroux’s correspondance gives, in particular, for every contact structure on a closed 3-manifold an adapted open book decomposition. Hence, it exists a Reeb vector that is tangent to the binding and transverse to the interior of the pages. For this vector field, each page is a Birkhoff section and the dynamics of the flow can be studied from the first return map. This correspondence is...

6. Dan Cristofaro-Gardiner (IAS and UC Santa Cruz): The Kapovich-Polterovich question

18/05/2021 16:30

The group of Hamiltonian diffeomorphisms of a symplectic manifold admits a remarkable bi-invariant metric, called Hofer’s metric. Many basic questions about the geometry of this metric remain open. For example, in 2006 Kapovich and Polterovich asked whether or not this group, in the case of the two-sphere, is quasi-isometric to the real line. I will explain joint work with Humilière and...

7. Álvaro del Pino Gómez (Universiteit Utrecht): Flexibility of distributions through convex integration.

19/05/2021 15:00

Building on the work of Nash on C1-isometric embeddings, Gromov devised a method, called convex integration, to construct and classify solutions of partial differential relations. For the scheme to work, one must assume that the relation in question is ample (and often open as well). The idea behind ampleness is that it allows us to start with a formal solution and add to it rapidly...

8. Urs Frauenfelder (Universität Augsburg): Frozen planet orbits.

19/05/2021 16:30

Frozen planet orbits are periodic orbits in the Helium atom, which play an important role in the semiclassical treatment of Helium. In the talk I discuss them from a mathematical point of view and explain how they are related to Hamiltonian delay equations.

9. Jonathan Bowden (Universität Regensburg): Open books, Bourgeois contact structures and their properties

20/05/2021 15:00

Twenty years ago Frederic Bourgeois introduced a construction of contact structures on the product of any contact manifold M with a 2-torus given a choice of compatible open book, whose existence was proven by Giroux-Mohsen. In particular, this yielded contact structures on all odd-dimensional tori answering a question of Lutz from the 70’s. A systematic study of these contact manifolds was...

10. Lev Buhovsky (Tel Aviv University): On Fabry's quotient theorem.

20/05/2021 16:30

The Fabry quotient theorem states that for a complex power series with unit radius of convergence, if the quotient of its consecutive coefficients tends to s, then the point z=s is a singular point of the series. In my talk I will try to describe an elementary proof of the theorem.

11. Russel Avdek (Uppsala Universitet): Holomorphic curve invariants of convex hypersurfaces.

21/05/2021 15:00

Let S be a convex hypersurface with neighborhood N(S) inside of some contact manifold. When dim(S)=2 the contact topology of N(S) is governed by simple closed curves on S. However, few tools are currently available to study N(S) when dim(S)>2. We provide such a tool which is applicable in any dimension by computing the sutured contact homology of N(S) in terms of linearized invariants of the...

12. Lisa Traynor (Bryn Mawr College): Legendrian Torus and Cable Links.

21/05/2021 16:30

Legendrian torus knots were classified by Etnyre and Honda. I will explain the classification of Legendrian torus links. In particular, I will describe restrictions on the Legendrian torus knots that can be realized as the components of a Legendrian torus link, and I will give examples of Legendrian torus links that cannot be destabilized even though they do not have maximal Thurston-Bennequin...

3. Coffee Break

4. Coffee Break and discussion