C⁰ aspects of symplectic geometry

TITLES AND ABSTRACTS:



Marie-Claude Arnaud: On the C¹ and C² convergence to weak KAM solutions

This is a joint work with Xifeng Su (Beijing Normal University). I will consider the action of a Hamiltonian flow on the set of  Lagrangian submanifolds. Focusing on Lagrangian graphs in a cotangent bundle, I will recall the Lax-Oleinik construction of a semi-group describing the evolution of these graphs for a Tonelli Hamiltonian and I will recall Fathi results on the C⁰ convergence of this semi-group. Then I will explain why this convergence is in fact C¹ and sometimes C² in a weak sense.


Barney Bramham: Orbit travel and symbolic dynamics for 3D Reeb flows using finite energy foliations

Suppose a non-degenerate Reeb flow of a closed contact 3-manifold admits a finite energy foliation F. The rigid leaves in F divide the space into regions A, B, C etc. To a typical Reeb trajectory one can associate an infinite sequence of letters which records the regions it visits. It turns out that this give rise to a Markov chain associated to a directed graph whose vertices are the rigid leaves. Questions about orbit travel, i.e. existence of Reeb trajectories that visit specified parts of the contact manifold, are notoriously complex and potentially interesting for the study of space missions, and one gets some course description of this through the connectivity of the associated graph, which in turn can be understood from a more local analysis. Another consequence is that if the associated Markov chain has positive topological entropy then so does the Reeb flow, while if the Markov chain has zero entropy then the Reeb flow essentially has a (typically disconnected) global Poincaré surface. This is joint work with Umberto Hryniewicz and Gerhard Knieper.


Daniel Cristofaro-Gardiner: Subleading asymptotics of ECH capacities

In previous work, Hutchings, Ramos and I studied the embedded contact homology (ECH) spectrum for any closed three-manifold with a contact form, and proved a "volume identity" showing that the leading order asymptotics recover the contact volume. I will explain recent joint work that sharpens this asymptotic formula by estimating the subleading term. The main technical point needed in our work is an improvement of a key spectral flow bound in Taubes' proof of the three-dimensional Weinstein conjecture; the main goal of my talk will be to explain the ideas that go into this improvement.  I will also discuss some possibilities for obtaining sharp asymptotics.


Başak Gürel: From pseudo-rotations to holomorphic curves

In this talk we will discuss new connections between pseudo-rotations and holomorphic curves, recently found in a joint work with Erman Cineli and Viktor Ginzburg.


Pazit Haim-Kislev: 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 a certain subadditivity property of the capacity of a general convex body, and we bound the systolic ratio for simplices in ℝ⁴.


Helmut Hofer: Feral Curves and Minimal Sets

Theories like Symplectic Field Theory use periodic orbits to build symplectic invariants for odd-dimensional manifolds with a stable Hamiltonian structure and symplectic cobordisms between them. Having a stable Hamiltonian structure is a rather strong condition, but one knows that without it, periodic orbits might not exist, i.e. the building blocks for the theory are gone. It is, of course, hard to believe that a symplectic cobordism suddenly ceases to have any meaningful symplectic properties. In this talk we present strong evidence that there is still a lot of structure which, interestingly, is related to important dynamical questions. A new class of so-called  feral (pseudoholomorphic) curves relates symplectic properties to more general closed invariant subsets. As one of the applications we answer a question raised by M. Herman during his 1998 ICM talk by showing that a compact regular Hamiltonian energy surface in ℝ⁴ has a proper closed invariant subset. This is joint work with Joel W. Fish.



Kei Irie: Equidistributed periodic orbits of C^∞-generic three-dimensional Reeb flows

We prove that, for a C^∞-generic contact form λ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to dλ. This is a quantitative refinement of the C^∞-generic density theorem for three-dimensional Reeb flows, which was previously proved by the speaker.
The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits "representing" ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.


Patrice Le Calvez: Homoclinic intersection for area preserving diffeomorphisms of surfaces

In a joint work with Martin Sambarino (UdelaR, Montevideo), we prove that if S is a smooth compact boundaryless orientable surface of genus g, furnished with a smooth area form ω and Diff ^r_ω(S), 1 ⩽ r ⩽ +∞,
is the space of C ^r diffeomorphisms of S   then generically in Diff ^r_ω(S), every hyperbolic periodic point has a transverse homoclinic intersection. Consequently the topological entropy is positive on an open and dense
set of Diff ^r_ω(S). The proof is divided in three steps

i) proving that under "explicit classical generic properties" the conclusion occurs if there is more than 2g-2 periodic points;

ii) proving that if these generic properties are satisfied and there is exactly 2g-2 periodic points, then there is power of f that is isotopic to the identity (at least if g ⩾ 2);

iii) proving that if f ∈ Diff ^r_ω(S) is isotopic to the identity, it can be perturbed to h ∈ Diff ^r_ω(S) with more
than 2g-2 periodic points.


Frédéric Le Roux: Hamiltonian Pseudo-rotations with non trivial dynamics

Hamiltonian pseudo-rotations are Hamiltonian diffeomorphisms with the minimal possible number of periodic points. Their dynamical properties have been studied recently on ℂPⁿ by Ginzburg and Gurel. Using Anosov-Katok method, we construct hamiltonian pseudo-rotations on toric manifolds, with the minimal possible number of ergodic measures. This is joint work with Sobhan Seyfaddini.


Marco Mazzucchelli: Min-max characterizations of Besse and Zoll Reeb flows

A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all periodic with the same minimal period. This class of manifolds has been thoroughly studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors. It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and only if its unit-speed periodic geodesics all have the same minimal period. In this talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds on the work of Lusternik and Schnirelmann. I will then present a stronger version of this statement valid for general Reeb flows on closed contact 3-manifolds: the closed orbits of any such Reeb flow admit a common period if and only if every orbit of the flow is closed. Time permitting, I will also summarize some related results for Reeb flows on higher dimensional contact spheres and for geodesic flows on simply connected compact rank-one symmetric spaces. The talk is based on joint works with Suhr, Cristofaro Gardiner, and Ginzburg-Gürel.


Stefan Müller: C⁰-characterization of symplectic via shape invariants

We prove that an embedding of a (small) ball into a symplectic manifold is symplectic if and only if it preserves the shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back of a fixed primitive of the symplectic form by a Lagrangian embedding of a fixed (closed) manifold and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods. The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). From that we derive a new proof of C⁰-rigidity of symplectic embeddings (and diffeomorphisms). Advantages of our techniques are that they avoid the cumbersome distinction between symplectic and anti-symplectic, and that they can be adapted to the contact setting (via coisotropic or pre-Lagrangian embeddings). The talk ends with a discussion of characterizations of other notions in symplectic geometry via shape invariants.


Kaoru Ono: Twisted sectors in Lagrangian Floer theory

After recalling some resuIts on anti-symplectic involution and Lagrangian Floer theory (based on joint work with Fukaya, Oh, Ohta) I will speak on a notion of twisted sectors in Lagrangian Floer theory in an appropriate setting for Lagrangian Floer theory.  It is based on a joint work (in progress) with Bohui Chen and Bai-Ling Wang.


Emmanuel Opshtein: Squeezing Lagrangian tori in ℝ⁴

A general framework for the talk may be called the "quantitative geometry" of Lagrangian submanifolds. I will consider here to which extent a standard Lagrangian torus in ℝ⁴ can be squeezed into a small ball. The result is quite surprising. When  the ratio between a and b is less than 2, the split torus T(a,b) is completely rigid, and cannot be squeezed into B(a+b) by any Hamiltonian isotopy. When this ratio exceeds 2 on the contrary, flexibility shows up, and T(a,b) can be squeezed into the ball B(3a).  This is a joint work with Richard Hind.


Vinicius Gripp Barros Ramos: A C⁰ version of the Arnold-Liouville theorem and symplectic embeddings

The Lagrangian bidisk is a 4-dimensional symplectic manifold with a surprisingly flexible embedding into a ball. This embedding is contructed first using the Arnold-Liouville theorem and techniques from ECH capacities. In this talk, I will explain how to extend this machinery to the L^p-sum of two disks, where different kinds of rigidity and flexibility appear. We will need a C⁰ version of the Arnold-Liouville theorem if 1<p<2. This is joint work with Yaron Ostrover.


Egor Shelukhin: TQFT operations and a conjecture of Viterbo

We explain how the TQFT operations introduced and studied by Seidel and Solomon in the context of symplectic cohomology and Lagrangian Floer homology yield upper bounds on the Lagrangian spectral norm of exact Lagrangians in cotangent disk bundles. These upper bounds in the case of the n-torus were conjectured by Viterbo. Time permitting, we discuss different TQFT operations in fixed point Floer homology introduced by Seidel, the speaker, and Zhao, and their application to questions in dynamics, such as the Hofer-Zehnder conjecture.


Jake Solomon: The space of positive Lagrangian submanifolds

It is a question of fundamental importance in symplectic geometry to determine when a Lagrangian submanifold can be moved by Hamiltonian flow to a volume minimizing submanifold. I will describe an approach to this problem based on the geometry of the space of positive Lagrangians. This space admits a Riemannian metric of non-positive curvature and a convex functional with critical points at volume minimizing Lagrangians. Existence of geodesics in the space of positive Lagrangians implies uniqueness of volume minimizing Lagrangians as well as rigidity of Lagrangian intersections. The geodesic equation is a degenerate elliptic fully non-linear PDE. I will discuss some results on the existence of solutions to this PDE and the relation to C⁰ symplectic geometry.


Alfonso Sorrentino: On the dynamics of conformally symplectic systems

In this talk I shall describe an extension of Aubry-Mather theory to a class of dissipative systems, namely the so-called conformally symplectic systems.


Shira Tanny: The Poisson bracket conjecture in dimension 2

I will discuss the Poisson bracket invariant of covers and explain how elementary geometric arguments can be used to prove Polterovich's pb-conjecture in dimension 2. This is a joint work with L. Buhovsky and A. Logunov, with a contribution of F. Nazarov.


Claude Viterbo: Barcodes and the spectrum of the Witten Laplacian: the degenerate case

We shall prove how the Barcodes gives information on asymptotic behaviour of the eigenvalues of the Witten laplacian associated to a function f on a smooth manifold, even in the case f is not Morse. This is joint work with F. Nier (Paris-Nord) and D. Le Peutrec (Orsay).


Jun Zhang: Stability results in symplectic geometry

In this talk, I will discuss various stability results and their applications to large-scale geometric properties of the spaces of geometric objects in symplectic geometry, for instance, Hamiltonian diffeomorphism groups or the space of star-shaped domains of Liouville manifolds. Stability results describe, in a quantitative manner, how the change of algebraic invariants controls the comparison of geometric objects. The central concept is called symplectic Banach-Mazur distance which can be viewed as a non-linear analog of the classical Banach-Mazur distance on convex bodies. This talk is mainly based on two joint work with Vukašin Stojisavljević and Richard Hind, respectively.







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