Jungsoo Kang (Seoul National University)
Title: Clarke duality and applications
Abstract: In contrast to the Hamiltonian action functional, Clarke’s dual action functional is defined only for convex Hamiltonian functions. However, it has better analytical properties. For example, the Morse homology of Clarke’s dual functional is well defined, and minimal periodic orbits arise as minimizers. In this mini-course, I will discuss such properties of Clarke’s dual functional, an isomorphism between the Morse homology of the dual functional and the Floer homology of the Hamiltonian functional, and some applications based on this isomorphism.
Vinicius Ramos (IMPA Rio de Janeiro)
Title: Billiards and Symplectic Topology
Abstract: In this mini-course, we will talk about billiard dynamics and the surprising link between the Mahler and the Viterbo conjectures. We will then explain how the integrability of many billiards can be used to prove that many domains in cotangent bundles are toric and how to use this to find very interesting sharp symplectic embeddings.
Urs Frauenfelder (Universität Augsburg)
Title: Growth of eigenvalues of Floer Hessians
Abstract: This is joint work with Joa Weber. When Floer introduced his celebrated homology he considered a gradient with respect to a very week metric. In particular, the Hessian for such a metric is not a operator from a fixed Hilbert space into itself but an operator on a scale of Hilbert spaces. We are looking into the topology of the space of Floer Hessians and show that this space has infinitely many connected components. This results is obtained by proving that the growth type of negative and positive eigenvalues are homotopy invariants.
Laurent Coté (Universität Bonn)
Title: Morse-Bott methods in Floer homotopy theory
Abstract: I will talk about a joint project with Y. Baris Kartal whose purpose is to incorporate Morse--Bott methods into Floer homotopy theory. Morse--Bott methods have the prospect of being useful for two reasons. First, they simplify equivariant constructions: for example, we used these methods to define a lift of circle equivariant symplectic homology to equivariant spectra. Second, they enable new computations. As a step in this direction, we used Morse--Bott methods to compute the (equivariant) local Floer homotopy of the orbit of an autonomous Hamiltonian, but I hope and expect that one can push such computations significantly further.
Oliver Edtmair (UC Berkeley)
Title: Systoles of convex energy hypersurfaces
Abstract: Hofer-Wysocki-Zehnder proved that every strictly convex energy hypersurface in R^4 possesses a disk-like global surface of section. They asked whether a systole, i.e. a periodic orbit of least action, must span such a disk-like global surface of section. In my talk, I will give an affirmative answer to this question. Moreover, I will discuss some implications of this result concerning normalized symplectic capacities. This is based on joint work in progress with Abbondandolo and Kang.
Yaron Ostrover (Tel Aviv University)
Title: Symplectic Barriers
Abstract: In a seminal 2001 paper, Biran introduced the concept of Lagrangian Barriers, describing a symplectic rigidity phenomenon characterized by obligatory intersections with Lagrangian submanifolds. In this talk, we discuss an analogous phenomenon for symplectic submanifolds, i.e., a rigidity stemming from mandatory intersections with symplectic submanifolds. The talk is based on joint work with Pazit Haim-Kislev and Richard Hind.
Marcelo Atallah (University of Sheffield)
TItle: The number of periodic points of surface symplectomorphisms
Abstract: A celebrated result of Franks shows that a Hamiltonian diffeomorphism of the sphere with more than two fixed points must have infinitely many periodic points. We present a symplectic variant of this phenomenon for symplectomorphisms of surfaces of higher genus that are isotopic to the identity; it implies an upper bound for the Floer-homological count of the number of fixed points of a symplectomorphism with finitely many periodic points. From a higher dimensional viewpoint, this can be understood as evidence for a non-Hamiltonian variant of Shelukhin’s result on the Hofer-Zehnder conjecture. Furthermore, we discuss the construction of a symplectic flow on a surface of any positive genus having a single fixed point and no other periodic orbits. This is joint work with Marta Batoréo and Brayan Ferreira.
Leonardo Macarini (IMPA Rio de Janeiro)
Title: Periodic orbits of non-degenerate lacunary contact forms on prequantizations
Abstract: A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of my knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. I will show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold $B$ has precisely $r_B$ contractible closed orbits, where $r_B=\dim H_*(B;{\mathbb Q})$. Examples of such prequantizations include the standard contact sphere and the unit cosphere bundle of a compact rank one symmetric space (CROSS). I will also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces. This is joint work with Miguel Abreu.
Pedro Salomão (SUSTech Shenzhen)
Title: Symplectic Dynamics and the Spatial Isosceles Three-Body Problem
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. The main results are joint with Xijun Hu (Shandong University), Lei Liu (Shandong University), Yuwei Ou (Shandong University) and Guowei Yu (Nankai University)
Vincent Colin (Université de Nantes)
Title: Reeb dynamics on sutured 3-manifolds
Abstract: In a joint work with Paolo Ghiggini and Ko Honda, we prove an isomorphism between the sutured versions of Heegaard Floer and Embedded contact homologies on a contact sutured 3-manifold. We derive from this a version of the Weinstein conjecture for contact 3-manifolds with sutured boundary, as well as other dynamical consequences.
Pierre Dehornoy (Aix Marseille Université)
Title: 3-dimensional flows, Birkhoff sections, and genus
Abstract: Birkhoff sections (sometimes called global surfaces of section) for a 3d-flow are a powerful tool from both topological and dynamical points of view. For transitive Anosov flows, their existence is known for 40 years, thanks to Fried. However the question of whether there always exists a genus-1 section is still open. I will present some results pointing toward a positive answer.
Vincent Humilière (IMJ-PRG Sorbonne Université)
Title: Morse/Floer theory with DG-coefficients and the almost sure existence property in cotangent bundles.
Abstract: In joint work with Jean-François Barraud, Mihai Damian and Alexandru Oancea we develop a Morse/Floer theory with coefficients in a DG-local system, and building on the work of Barraud and Cornea (2004). This allows for instance to recover the homology of a fibration (whose fiber does not necessarily have finite dimension) from Morse/Floer data on the base of the fibration. I will present an application to the almost sure existence property of a periodic orbit in cotangent bundles.
Daniel Cristofaro-Gardiner (University of Maryland, College Park)
Title: Low-action holomorphic curves and invariant sets
Abstract: I will discuss a new compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy; in the limit, we extract a family of closed Reeb-invariant subsets. I will also explain why such sequences exist in abundance in low-dimensional symplectic dynamics, via the theory of embedded contact homology. This has various applications: the one I want to focus on in my talk is a very general Le Calvez-Yoccoz type property. All of this is joint with Rohil Prasad.
Gabriele Benedetti (Vrije Universiteit Amsterdam)
Title: Periodic magnetic geodesics with low energy in high dimension
Abstract: We discuss the existence of periodic magnetic geodesics on low energy levels for manifolds of dimension at least four. To get the necessary compactness in the variational setting, we combine Thom’s transversality theorem with index estimates coming from the positivity of the magnetic Ricci curvature. This is very much work in progress, joint with Valerio Assenza.
Sheila Sandon (Université de Strasbourg)
Title: Contact non-squeezing and translated chains
Abstract: In 2006 Eliashberg, Kim and Polterovich proved, using SFT techniques, that for any integer k there is no contact isotopy of R^2n x S^1 sending the prequantization of a ball of capacity bigger than k to the prequantization of a ball of capacity smaller than k. This result was extended to the case of balls of capacity bigger than 1 (not necessarily separated by an integer) by Chiu in 2017 and Fraser in 2016, using respectively microlocal sheaves and SFT. In my talk I will present a proof of this general contact non-squeezing theorem using generating functions. I'll focus in particular on the key role played in the proof by translated chains of contactomorphisms, a generalization of translated points. This is a join work with Maia Fraser and Bingyu Zhang.
Marco Mazzucchelli (ENS de Lyon)
Title: From curve shortening to flat link stability and Birkhoff sections of geodesic flows
Abstract: In this talk, based on joint work with Marcelo Alves, I will present three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is the stability, under C^0 -small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic \gamma forces the existence of infinitely many closed geodesics in every primitive free homotopy class of loops and intersecting \gamma. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface of positive genus.
Bernhard Albach (RWTH Aachen)
Title: On the number of closed geodesics on S^2
Abstract: Given an arbitrary metric on S^2, the question of how many closed godesics this metric has has a long history. In this talk, we will present a result stating that the number of closed geodesics grows quadratically with respect to length. The proof of this is based on methods from cylindrical contact homology, dynamics of annulus maps and spheres of revolution.
Zhen Gao (Universität Augsburg)
Title: Double the rectangular pegs
Abstract: In this talk, I will report on recent work (arXiv:2404.13209) on the multiplicity result of rectangular pegs based on the a prior existence result by Greene and Lobb. The proof involves computing the Euler characteristic of Lagrangian Floer homology. We obtain a convenient formula for computing the algebraic intersection number of cleanly intersecting Lagrangian submanifolds, which is well-consistent with the Euler characteristic of Floer homology in the spirit of ``categorification''. If time permits, I will discuss possible tasks for showing the a prior existence of rectangular pegs via Floer theory.
Hanna Häußler (Universität Augsburg)
Title: Generalisation of Arnolds J+ invariant for pairs of immersions
Abstract: Arnold’s J+ invariant is used to study periodic orbits of physical systems in the plane. It is sensitive to direct self-tangency during homotopy and thus as in solution of ODE’s such a move is not allowed, marks families of periodic orbits. In this talk we will widen this approach and introduce a J+-like invariant for pairs of immersions, called the J2+-invariant, which has several nice properties and gives new opportunities to get a better understanding of concrete physical systems, like the rotating Kepler problem or the Euler problem.
Leonardo Masci (RWTH Aachen)
Title: A Poincaré-Birkhoff theorem for asymptotically linear Hamiltonian diffeomorphisms
Abstract: The celebrated Poincaré-Birkhoff theorem on area-preserving maps of the annulus is of fundamental importance in the fields of Hamiltonian dynamics and symplectic topology. In this talk I will formulate a twist condition, inspired by the Poincaré-Birkhoff theorem, which applies to the asymptotically linear Hamiltonian diffeomorphisms of Amann, Conley and Zehnder. When this twist condition is satisfied, together with some technical assumptions, the existence of infinitely many periodic points is obtained.
Stefan Matijevic (RUB Bochum)
Title: Positive (S1-equivariant) symplectic homology of convex domains, Gutt–Hutchings capacities, and Clarke's duality.
Abstract: We show that the positive (S1-equivariant) symplectic homology of a convex domain is naturally isomorphic to the singular (S1-equivariant) homology of Clarke's dual associated with the convex domain. Consequently, we show that Gutt–Hutchings capacities coincide with the spectral invariants introduced by Ekeland-Hofer on the set of convex domains. Also, as a corollary, we get that barcode entropy associated with the singular homology of Clarke's dual is a lower bound for the topological entropy of the Reeb flow on the boundary of a convex domain in R^2n. In particular, the above-mentioned barcode entropy coincides with the topological entropy for convex domains in R4.
Francesco Morabito (École Polytechnique - Paris)
Title: A filtration in linking numbers for generating functions
Abstract: In this talk we focus our attention on compactly supported Hamiltonian diffeomorphisms of the disc. To each pair of distinct fixed points we can associate an integer, the linking number of the pair. Moreover, we can describe the dynamics of the diffeomorphism itself via the Morse complex of a generating function. This Morse complex is classically known to be filtered by action. In this talk we are going to show how we can equip its tensor power with a secondary filtration which keeps track of the linking numbers of pairs of orbits. The proof relies on foundational work on twist maps carried out by Patrice Le Calvez in the '90s, and on uniqueness properties of generating functions. If time permits, we are going to give some ideas of an analogue filtration one can construct on the tensor power of the Hamiltonian Floer complex.
Simon Vialaret (RUB Bochum)
Title: Systolic inequalities for S1-invariant contact forms
Abstract: In contact geometry, a systolic inequality aims to give a uniform upper bound on the shortest period of a closed Reeb orbit, in term of the contact volume. Although it is known that there is no systolic inequality for general contact forms on a given contact manifold, I will state a systolic inequality valid for contact forms that are invariant under a circle action in dimension 3.