Department of Mathematics

Abstracts and slides

Frédéric Bourgeois 
Université Paris-Sud, France 

Title: 

Reeb dynamics and contact handle attachments

Abstract: 

In this ongoing joint work with Anne Vaugon, we provide a suitable
framework to study the attachment of a handle to a contact
manifold with boundary. Any closed contact manifold can be
constructed via successive attachments of such handles.
We then describe the creation of periodic Reeb orbits by
this operation, in view of understanding its effect on contact
invariants such as sutured contact homology. Similarities and
differences with handle attachments on Weinstein manifolds
will be underlined.  


Jean Gutt 
University of Cologne, Germany 

Title: 

Non-equivalent symplectic embeddings

Abstract: 

I will discuss a joint result with Mike Usher, showing that many toric domains X in the 4-dimensional euclidean space admit symplectic embeddings f into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes f(X) to X.


Sonja Hohloch 
University of Antwerpen, Belgium 

Title: 

Compact semitoric systems with 2 focus-focus singularities and the twisting index

Abstract: 

A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poissoncommute and induce an (S1 × R)-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo & Vu Ngoc [PVG1, PVG2] by means of five invariants. One of these five invariants is the so-called twisting index which compares the ‘distinguished’ torus action given near each focus-focus singular fiber to the global toric ‘background action’. Although the abstract definition of the twisting index is not very difficult, it has never been abstractly studied or calculated for explicit examples. One of the reasons may have been that there were no known explicit examples of semitoric systems having more than one focus-focus singularity. In this talk, we

• give various geometric interpretations of the twisting index;

• present a family of compact semitoric systems having two focusfocus points, cf. [HP]. This is a joint project with Joseph Palmer (Rutgers University). We hope that one of the twisting index reformulations will allow us in the future to compute the twisting index for this family of systems.

References

[HP] Hohloch, S.; Palmer, J.: A family of compact semitoric systems with two focus-focus singularities. arXiv:1710.05746, 27p.
[PVG1] Pelayo, A.; Vu Ngoc, S.: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math., 177 (3), 571 – 597, 2009.
[PVG2] Pelayo, A.; Vu Ngoc, S.: Constructing integrable systems of semitoric type. Acta Math., 206 (1), 93 – 125, 2011.


Cecilia Karlsson 
Oslo University, Norway 

Title: 

Orientations of moduli spaces for Legendrian contact homology

Abstract: 

In this talk, I will discuss how moduli spaces of Morse flow trees in Legendrian contact homology (LCH) can be oriented in a coherent and computable manner, obtaining a Morse-theoretic way to compute LCH with integer coefficients. This is built on the machinery of capping disks, and I will briefly explain how different systems of capping disks affect the orientations. This, in turn, uses the fact that an exact Lagrangian cobordism with cylindrical Legendrian ends induces a morphism between the LCH-complexes of the ends, which can be proven to hold also with integer coefficients.  


Oleg Lazarev 
Columbia University, USA 

Title: 

Simplifying Weinstein Morse functions

Abstract: 

By work of Cieliebak and Eliashberg, any Weinstein structure on Euclidean space that is not symplectomorphic to the standard symplectic structure necessarily has at least three critical points; an infinite collection of such exotic examples were constructed by McLean. I will explain how to use handle-slides and loose Legendrians to show that this lower bound on the number of critical points is exact; that is, any Weinstein structure on Euclidean space R2n has a compatible Weinstein Morse function with at most three critical points (of index 0, n-1, and n). Furthermore, the number of gradient trajectories between the index n-1, n critical points can be uniformly bounded independent of the Weinstein structure. Similarly, any Weinstein structure on the cotangent bundle of the sphere of dimension at least 3 has a compatible Weinstein Morse function with two critical points. As applications, I will give new proofs of some existing h-principles and present some new constructions of exotic cotangent bundles. 


Burak Özbağcı 
Koç University, Turkey 

Title: 

Fillings of unit cotangent bundles of nonorientable surfaces

Abstract: 

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed connected smooth surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)  


Daniel Pomerleano 
Imperial College, UK 

Title: 

Mirror symmetry and symplectic cohomology

Abstract: 

I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Stanley-Reisner ring on the dual intersection complex of a compactifying divisor. I will explain how this result relates to work of Gross, Hacking, Keel, and Siebert.  


Silvia Sabatini 
University of Cologne, Germany 

Title: 

Hamiltonian S1-spaces with large minimal Chern number

Slides:

PDF

Abstract: 

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 concentrare 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.

References

[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).


Felix Schmäschke 
Humboldt University, Germany 

Title: 

Abelianization and quantum cohomology of symplectic quotients

Abstract: 

We consider closed monotone symplectic manifolds obtained by symplectic reduction with respect to a nonabelian compact Lie group. We show that if twice the minimal Chern number of the quotient is strictly larger than the dimension minus the rank of the group, then an isomorphism suggested by Martin holds for quantum cohomology. This isomorphism relates the quantum cohomologies of quotients by the group and by its maximal torus. As an application we give a concrete presentation of the quantum cohomology ring of the symplectic quotient in the case of a linear group action.  


Nick Sheridan 
University of Camridge, UK 

Title: 

Symplectic mapping class groups and homological mirror symmetry

Abstract: 

I will explain how one can get new information about symplectic mapping class groups by combining two recent results: a proof of homological mirror symmetry for a new collection of K3 surfaces (joint work with Ivan Smith), together with the computation of the derived autoequivalence group of a K3 surface of Picard rank one (Bayer—Bridgeland). For example, it is possible to give an example of a symplectic K3 whose smoothly trivial symplectic mapping class group (the group of isotopy classes of symplectic automorphisms which are smoothly isotopic to the identity) is infinitely-generated. This is joint work with Ivan Smith.  


Vukašin Stojisavljević
Tel Aviv University, Israel 

Title: 

Persistence modules with operators

Slides:
 
PDF

Abstract: 

Originating in topological data analysis, persistence modules provide
an algebraic formalism which can be used to systematically study
filtered homologies and obtain robust numerical invariants. The theory
has recently found applications in symplectic geometry in the work of
L. Polterovich and E. Shelukhin, where filtered Floer homology was
treated as a persistence module. An operator on a (graded) persistence module V is a morphism from V to itself which shifts the filtration (and grading) by a fixed constant. Examples of operators on filtered Morse and Floer homology are given by intersection and quantum product with a fixed homology class. We will introduce the general framework of persistence modules with operators and present some applications to Hofer's geometry and C0-geometry of Morse functions. The talk is based on a joint work with L. Polterovich and E. Shelukhin.