Convexity in contact and symplectic topology

4 juil. 2022, 09:00 → 8 juil. 2022, 18:00 Europe/Paris

Amphithéâtre Hermite (Institut Henri Poincaré)

Picture © by Steve Murez 2022

This event, initially scheduled in 2021, will celebrate the 60th birthday of Emmanuel Giroux. It will happen in Institut Henri Poincaré in Paris, there is ordinary practical information on the IHP website and Covid related information on the relevant ministry website.

Talks will be lived streamed here.

Speakers

• Mohammed Abouzaid (Columbia University)
• Daniel Alvarez-Gavela (MIT)
• Mélanie Bertelson (Université Libre de Bruxelles)
• Paul Biran (ETH Zürich)
• Yasha Eliashberg (Stanford University)
• Paolo Ghiggini (Nantes Université)
• Ko Honda (University of California, Los Angeles)
• Ailsa Keating (University of Cambridge)
• Oleg Lazarev (University of Massachusetts Boston)
• Noémie Legout (Uppsala University)
• Jean-Paul Mohsen (Aix-Marseille Université)
• Yu Pan (Tianjin University)
• Ana Rechtman (Université de Strasbourg)
• Kyler Siegel (University of Southern California)
• Ivan Smith (University of Cambridge)
• András Stipsicz (Rényi Institute of Mathematics)
• Anne Vaugon (Université Paris-Saclay)
• Morgan Weiler (Cornell university)

Organizers

• Vincent Colin
• Sylvain Courte
• Hélène Eynard-Bontemps
• Patrick Massot
• Sobhan Seyfaddini

 

 

lundi 4 juillet

1 Contact convexity from Giroux to Honda-Huang

Contact convexity theory which was pioneered 30 years ago by Emmanuel Giroux played and continue to play an important role in contact and symplectic geometry.
More recently, Ko Honda and Yang Huang discovered new surprising flexibility phenomena in the high dimensional contact convexity.
In the talk I will discuss a simplified approach (joint with Dishant Pancholi) to Honda-Huang’s results.

Orateur: Yakov Eliashberg (Stanford university)

2 A h-principle for conformal symplectic structures

Any non-degenerate $2$-form can be homotoped to a locally conformal symplectic structure whose Lee form can be chosen to be any non-vanishing closed $1$-form. Each component of the boundary can be chosen to be concave or convex and to inherit a given overtwisted contact structure. On the other hand, for codimension one foliations, a leafwise conformal symplectic structure whose Lee form coincides with the holonomy $1$-form yields a contact structure. Unfortunately, the h-principle described above does not admit a foliated version unless the ambient manifold has a non-empty boundary. This is a joint work with Gaël Meigniez.

Orateur: Mélanie Bertelson (Université Libre de Bruxelles)

3 Limits in Donaldson-Auroux theory

In the mid-90's, Donaldson has introduced asymptotically holomorphic techniques in symplectic geometry. In this talk, I will discuss some applications of Donaldson's construction and I will present a reformulation of the results. The (renormalized) limits are the main tool in this reformulation.

Orateur: Jean-Paul Mohsen (Aix-Marseille Université)

4 Complex cobordism and loops of Hamiltonians

I will describe joint work with McLean and Smith showing that loops of Hamiltonians are trivial from the point of view of complex cobordism. The proof is a surprising application of Floer homology theory, and of abstract results in chromatic homotopy theory.

Orateur: Mohammed Abouzaid (Columbia University)

mardi 5 juillet

5 New constructions of symplectomorphisms

We introduce two new constructions of compactly supported symplectomorphisms of Weinstein 4-manifolds: Lagrangian translations' andnodal slide recombinations'. These are natural from the perspective of mirror symmetry. After an overview of the constructions and their properties, the talk will focus on describing the maps in the first non-trivial cases. Joint work with Paul Hacking.

Orateur: Ailsa Keating (University of Cambridge)

6 K-theoretic aspects of the nearby Lagrangian conjecture

It was recently shown by M. Abouzaid, S. Courte, S. Guillermou and T. Kragh that every nearby Lagrangian admits a so-called twisted generating function of tube type, thereby establishing a connection between the nearby Lagrangian conjecture and Waldhausen's algebraic K-theory of spaces. I will discuss several aspects of this connection, including a joint work in progress with M. Abouzaid, S. Courte and T. Kragh which finds new restrictions on the smooth structure of nearby Lagrangians.

Orateur: Daniel Álvarez-Gavela (MIT)

7 Almost all Reeb vector fields admit a Birkhoff section

A Birkhoff section reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism and provides a rational open book carrying the flow. I will present a recent existence statement for Birkhoff sections: the set of a Reeb vector fields on closed 3-manifolds  that admit a Birkhoff section contains an open and dense subset in the $C^\infty$ topology. This construction is based on the existence of broken book decompositions and is part of a joint work with Vincent Colin, Pierre Dehornoy and Umberto Hryniewicz.

Orateur: Ana Rechtman (Université de Strasbourg)

8 Singular plane curves and stable nonsqueezing phenomena

The existence of rational plane curves of a given degree with prescribed singularities is a subtle and active area in algebraic geometry. This question turns out to be closely related to difficult enumerative problems which arise in symplectic field theory, and which in turn play a key role in the theory of high dimensional symplectic embeddings. In this talk I will discuss various perspectives on these enumerative problems and how recent advances on the symplectic side can give insight into the theory of singular curves and vice versa.

Orateur: Kyler Siegel (University of Southern California)

mercredi 6 juillet

9 Surfaces in smooth 4-manifolds

After reviewing methods for constructing exotic smooth structures on closed four-manifolds, we examine the ‘genus-function’ on the second homology, and ask/answer some questions related to this function. We extend the notion to manifolds with boundary, where the surfaces are bounded by knots or links in the boundary. We examine the relevance of these notions for the Smooth Four-dimensional Poincare Conjecture.

Orateur: András Stipsicz (Renyi institute)

10 Floer homology for singular lagrangians and homological mirror symmetry of CP^n

I will explain how to define a version of Floer homology for Lagrangians with conical singularities and how, in good situations, this construction leads to the definition of localised mirror functors which generalise those of Cho-Hong-Lau. Then I will apply this construction to find the mirror of the pair (CP^n, D) where D={x_0=0} \cup {x_1...x_n=x_0^n }. This is a joint work in progress with Georgios Dimitroglou Rizell.

Orateur: Paolo Ghiggini (Nantes Université)

11 Foulon-Hasselblatt contact surgery and orbit growth of Reeb flows

This talk will focus on dynamical properties of Reeb vector fields after a Legendrian surgery. Our description of the surgery was originally conceived by Foulon and Hasselblatt as a source of Anosov Reeb flows on various 3-manifolds including hyperbolic examples. I will explain that this operation often increases the complexity of Reeb flows by studying their orbit growths. This talk is based on joint works with B. Hasselblatt and P. Foulon and with S. Tapie.

Orateur: Anne Vaugon (Université Paris-Saclay)

12 Area-preserving homeomorphisms and link spectral invariants

We will discuss various results concerning the algebraic structure of the group of area-preserving homeomorphisms of a compact surface. The results are obtained from the asymptotics of spectral invariants associated to configurations of disjoint circles on the surface. These link spectral invariants are in turn defined from the Floer cohomology of an associated Lagrangian in the symmetric product. This talk reports on joint work with Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk-Yu Mak and Sobhan Seyfaddini.

Orateur: Ivan Smith (University of Cambridge)

13 Reception at ENS

jeudi 7 juillet

14 SFT-style Rabinowitz complex for exact Lagrangian cobordisms and Calabi-Yau isomorphism

We will define a Floer complex (the "Rabinowitz" complex) associated to a pair of exact Lagrangian cobordisms, using SFT techniques. This complex is a DG-bimodule over the Chekanov-Eliashberg algebras of the Legendrian submanifolds in the negative end of the cobordisms. We will use this complex and its properties to show that the Chekanov-Eliashberg algebra of an horizontally displaceable Legendrian sphere satisfies some Calabi-Yau property, namely that it is quasi-isomorphic as a DG-bimodule over itself to its inverse dualizing bimodule.

Orateur: Noémie Legout (Uppsala University)

15 ECH capacities and fractals of infinite staircases of 4D symplectic embeddings

The ellipsoid embedding function of a symplectic manifold measures the amount by which the symplectic form must be scaled in order to fit an ellipsoid of a given eccentricity. It generalizes the Gromov width and ball packing numbers. In 2012 McDuff and Schlenk computed the ellipsoid embedding function of the ball, showing that it exhibits a delicate piecewise linear pattern known as an infinite staircase. Since then, the embedding function of many other symplectic four-manifolds have been studied, and not all have infinite staircases. We will classify those symplectic Hirzebruch surfaces whose embedding functions have an infinite staircase, and explain how our work provides a blueprint for other targets. Based on work with Magill and McDuff and work in progress with Magill and Pires.

Orateur: Morgan Weiler (Cornell university)

16 Contact structures and open book decompositions

Around twenty years ago Emmanuel Giroux formulated the equivalence of contact structures and open book decompositions with Weinstein pages up to stabilization. We revisit this equivalence through the lens of more recent developments in convex hypersurface theory.
This is joint work with Joe Breen and Yang Huang.

Orateur: Ko Honda (University of California, Los Angeles)

vendredi 8 juillet

17 Reverse Lagrangian surgery on fillings

For an immersed filling of a topological knot, one can do surgery to resolve a double point with the price of increasing surface genus by 1. In the Lagrangian analog, one can do Lagrangian surgery on immersed Lagrangian fillings to treat a double point by a genus. In this talk, we will show that not all Lagrangian surgeries are reversible. Moreover, there are surgeries that can not be reversed in the Lagrangian world but  are potentially able to be reversed in the smooth world.

Orateur: Yu Pan (Tianjin University)

18 Localization and flexibilization in symplectic geometry

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data)  become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (away from zero ). Building on work of Abouzaid and Seidel, l will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals) that interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogs of topological localization of Sullivan, Quillen, and Bousfield. 
This talk is based on joint work with Z. Sylvan and H. Tanaka.

Orateur: Oleg Lazarev (University of Massachusetts Boston)

19 Persistence K-theory

K-theory, in its classical form, associates to a triangulated category an abelian group called the K-group (or the Grothendieck group). Important invariants of various triangulated
categories are known to factor through their K-groups.

In this talk we will explain the foundations of persistence K-theory, which is an analogous theory for triangulated persistence categories. In particular we will introduce new persistence measurements coming from these K-groups, and new invariants coming from the combination of the persistence and triangulated structures.

In the last part of the talk we will exemplify this new theory on the case of the persistence Fukaya category of Lagrangian submanifolds. In particular we will show how our invariants can distinguish between modules that can represent embedded Lagrangians and those who can
represent only immersed ones.

Based on joint work with Octav Cornea and Jun Zhang.

Orateur: Paul Biran (ETH Zürich)