Special Session 69: New developments in symplectic dynamics

On the minimal number of closed geodesics on positively-curved spheres

Huagui Duan Nankai University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will introduce the problem about the optimal number of closed geodesics on spheres. Recently it has been proved that for every Finsler metric on certain positively-curved spheres of dimension $n$, there exist at least $n$ prime closed geodesics, which solved a conjecture of Katok and Anosov for such spheres when $n$ is even, which is a joint work with Dong Xie.

Symplectic packing stability

Oliver Edtmair ETH Zurich Austria Co-Author(s):

Abstract: Biran proved that the full volume of a rational closed symplectic 4-manifold can be symplectically packed by any sufficiently large number of equally sized balls, a phenomenon known as packing stability. In my talk, I will explain how to extend this result to arbitrary compact symplectic 4-manifolds with smooth boundary. This is in contrast to a recent result of Cristofaro-Gardiner and Hind concerning the failure of packing stability in the case of non-smooth boundary. I will also explain how packing stability relates to the subleading asymptotics in the Weyl laws for ECH and PFH spectral invariants and to the simplicity/non-simplicity of groups of Hamiltonian diffeomorphisms/homeomorphisms.

Spectrally-large scale geometry and symplectic squeezing in cotangent bundle of torus

Qi Feng University of Science and Technology of China Peoples Rep of China Co-Author(s):    Jun Zhang

Abstract: This talk covers two separate topics. One, we identify a class of closed Riemannian manifolds for which there exists a rank-$\infty$ quasi-flat in the metric space of the Hamiltonian deformations of a fiber in the unit codisk bundle with respect to the Lagrangian spectral norm. Two, we prove that for $n\geq 2$ any subbundle of $T^* T^n$ with bounded fibers symplectically embeds into a trivial subbundle where the fiber is an irrational cylinder.

Degenerate Arnol`d conjectures, Hamiltonian periodic orbits and Lagrangian intersections

Wenmin Gong Beijing Normal University Peoples Rep of China Co-Author(s):

Abstract: Symplectic geometry has some wide open problems concerning Hamiltonian periodic orbits and Lagrangian submanifolds. Among them, Arnold`s conjectures are of central importance. In this talk, I will survey some important solved and unsolved problems about the degenerate homological Arnold conjecture.

The shape invariant of toric domains

Richard Hind University of Notre Dame USA Co-Author(s):

Abstract: We define a shape invariant, a sort of set valued symplectic capacity, for domains in $\mathbb{R}^4$. The shape of a domain $X$ captures the Hamiltonian isotopy classes, in $\mathbb{R}^4$, of embedded Lagrangian tori in $X$. Then we describe computations for a class of toric $X$, showing that the moment image and the shape coincide in certain regions. Hence we have rigidity for many symplectic embedding problems. This reports on joint work with Ely Kerman and Jun Zhang.

Rabinowitz Floer homology for prequantization bundles

Jungsoo Kang Seoul National University Korea Co-Author(s):

Abstract: Prequantization bundles are circle bundles over integral symplectic manifolds and come with canonical contact structures. In this talk, I will explain the construction of a Floer Gysin exact sequence for prequantization bundles using Rabinowitz Floer homology. Some applications will also be discussed. This is a joint work with Joonghyun Bae and Sungho Kim.

Lagrangian link quasimorphisms and the non-simplicity of Hameomorphism group of surfaces

Cheuk Yu Mak University of Sheffield England Co-Author(s):    Daniel Cristofaro-Gardiner, Vincent Humiliere, Sobhan Seyfaddini, Ivan Smith and Ibrahim Trifa

Abstract: In this talk, we will explain the construction of a sequence of homogeneous quasi-morphisms of the area-preserving homeomorphism group of the sphere using Lagrangian Floer theory for links. This sequence of quasi-morphisms has asymptotically vanishing defects, so it is asymptotically a homomorphism. It enables us to show that the Hameomorphism group is not the smallest normal subgroup of the area-preserving homeomorphism group. If time permits, we will explain how to generalize it to all positive genus surfaces even though we no longer have quasi-morphisms. The case of the sphere is joint work with Daniel Cristofaro-Gardiner, Vincent Humiliere, Sobhan Seyfaddini, and Ivan Smith. The case of positive genus surfaces is joint work with Ibrahim Trifa.

Barcode entropy of Lagrangian submanifolds

Matthias Meiwes Tel Aviv University Israel Co-Author(s):

Abstract: Recently, some fruitful approaches were found to express complexity of Hamiltonian or Reeb dynamics in terms of Floer theory or contact homology. One goes back to Alves and Pirnapasov, where the growth of essential homotopy classes of orbits in a complement of a link of periodic orbits in 3-dimensional flows is studied, another is due to Cineli, Ginzburg, and Gurel, through the study of barcode entropy. In my talk, I will explain some recent results on estimates of the barcode entropy of Hamiltonian diffeomorphisms relative to pairs of Lagrangian submanifolds, and discuss connections to other properties of the dynamics. Some of the results reveal moreover some interesting connections between the two approaches to complexity mentioned above.

Symplectic Dynamics and the Spatial Isosceles Three-Body Problem

Pedro A Salomao SUSTech Peoples Rep of China Co-Author(s):    Xijun Hu (Shandong University), Lei Liu (Shandong University), Ywei Ou (Shandong University), Guowei Yu (Nankai University)

Abstract: In this talk, I will discuss tools from Symplectic Dynamics that can be used to answer some questions in the spatial isosceles three-body problem. The main results are related to periodic orbits, global surfaces of section, and, more generally, transverse foliations. We focus on the dynamics of energy surfaces and explain how those objects change as the energy increases.

Symplectic classification of compact almost-toric systems of dimension four

Xiudi Tang Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: Almost-toric systems are important in mirror symmetry. We give a classification of $4$-dimensional compact almost-toric systems up to fiber-preserving symplecctomorphisms. This generalizes the classification by Pelayo--V\~u Ng\d{o}c on simple semitoric systems and that by the speaker together with Pelayo and Palmer on semitoric systems, both in dimension four. The extra difficulty for almost-toric systems is the lack of a global circle action. The polygon invariant is replaced by an almost-toric closed disk, and we give appropriate notions of focus-focus label and twisting indices in the almost-toric case.

Dynamics of composite symplectic Dehn twists

Jinxin Xue Tsinghua University Peoples Rep of China Co-Author(s):    Wenmin Gong, Zhijing Wang

Abstract: We show that composite symplectic Dehn twists have certain form of nonuniform hyperbolicity: it has positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, which are analogous to signatures of pseudo Anosov mapping classes. Moreover, we show that the rank of the Floer cohomology group of these compositions grows exponentially under iterations, which partially answers a question of Smith concerning the classification of symplectic mapping class group in higher dimensions. Finally, we propose a conjecture on the positive metric entropy of our model and point out its relationship with the standard map.

Givental`s non-linear Maslov index via Floer cones

Jun ZHANG University of Science and Technology of China Peoples Rep of China Co-Author(s):    Dylan Cant, Eric Kilgore, Igor Uljarevi\`{c}

Abstract: In this talk, we will present how to apply a recently developed Floer theory on a fillable contact manifold, called the contact Hamiltonian Floer homology, to generate a homological machinery that replaces the classical Givental non-linear Maslov indices. As a key step, we will emphasize the role of the homological mapping cone from this Floer theory (called a Floer cone) and its fundamental role in reflecting local data of periodic orbits. As an application, the multiplicity of translated points, serving as a natural generalization of fixed points in contact Hamiltonian dynamics, will be deduced. This talk is based on joint work with Dylan Cant, Eric Kilgore, and Igor Uljarevic.

Symplectic camel herd

Zhengyi Zhou Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Weiwei Wu

Abstract: I will explain the analog of the symplectic camel theorem with a coisotropic needle, i.e. non-triviality of certain higher loops of ball embeddings into an exotic symplectic R^{2n}. Joint work with Weiwei Wu.