Alberto Abbondandolo (Bochum University)
Title: Zoll contact forms are local maximizers of the systolic ratio
Abstract: The systolic ratio of a contact form on a closed (2n+1)-manifold is the ratio between the n-th power of the smallest period of closed Reeb orbits and the contact volume. A contact form is said to be Zoll if all the orbits of the corresponding Reeb flow are closed and have the same minimal period. I will discuss the proof of the theorem that is stated in the title (joint work with Gabriele Benedetti). This theorem implies local systolic inequalities in metric geometry, a local version of Viterbo’s conjecture about the symplectic capacity of convex bodies and a non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.
Alex Fauck (HU Berlin)
Title: Computing RFH for tentacular hyperboloids
Abstract: In my talk, I will compute the Rabinowitz-Floer homologies of tentacular hyperboloids by relating them to the RFH of the standard spheres. These hyperboloids are non-compact contact hypersurfaces and in contrast to the spheres, their RFH does not vanish and is not equal to their sigular homologies. This proves at the same time, that the Weinstein conjecture holds for these hyperboloids. This is joint work with W. Merry and J. Wiśniewska.
Urs Frauenfelder (Augsburg University)
Title: Frozen planet Orbits
Abstract: Frozen planet orbits are periodic orbits in the Helium atom, which play an important role in the semiclassical treatment of Helium. In the talk I discuss them from a mathematical point of view and explain how they are related to Hamiltonian delay equations.
Eva Miranda (Universitat Politècnica de Catalunya)
Title: Singular Floer theory and singular Hamiltonian/Reeb Dynamics: First steps
Abstract: Non-compact symplectic manifolds can sometimes be compactified as singular symplectic manifolds where the symplectic form "blows up" along an hypersurface in a controlled way (b^m-symplectic manifolds). In natural examples in Celestial mechanics such as the 3-body problem these compactifications are given by regularization transformations à la Moser/Mc Gehee etc. I will use the theory of b^m-symplectic/b^m-contact manifolds as a guinea pig to propose ways to extend Floer theory and the study of Hamiltonian/Reeb Dynamics to singular symplectic/contact manifolds. This, in particular, yields new results for non-compact symplectic manifolds and for special (but, yet, meaningful) classes of Poisson manifolds.
Inspiration comes from several results extending the Weinstein conjecture to the context of b^m-contact manifolds (joint with Cédric Oms) and its connection to the study of escape orbits in Celestial mechanics and Fluid Dynamis (joint with Cédric Oms and Daniel Peralta-Salas). Those examples motivate a model for (singular) Floer homology (work in progress with Joaquim Brugués and Cédric Oms).
I'll describe the motivating examples/results and some ideas to attack the general questions.
Agustin Moreno (Uppsala University)
Title: Contact geometry of the spatial restricted three-body problem
Abstract: In his search for closed orbits in the planar restricted three-body problem, Poincaré’s approach roughly reduces to:
(1) Finding a global surface of section;
(2) Proving a fixed-point theorem for the resulting return map.
This is the setting for the celebrated Poincaré-Birkhoff theorem. In this talk, I will discuss a generalization of this program to the spatial problem.
For the first step, we obtain the existence of global hypersurfaces of section for which the return maps are Hamiltonian, valid for energies below the first critical value and all mass ratios. For the second, we prove a higher-dimensional version of the Poincaré-Birkhoff theorem, which gives infinitely many orbits of arbitrary large period, provided a suitable twist condition is satisfied. We also present a construction that associates a Reeb dynamics on a moduli space of holomorphic curves, to the given dynamics. This is joint work with Otto van Koert.
John Pardon (Princeton University)
Title: Wrapped Fukaya categories and microlocal sheaves
Abstract: I will discuss joint work with Sheel Ganatra and Vivek Shende which establishes an equivalence, conjectured by Kontsevich and Nadler, between wrapped Fukaya categories and categories of microlocal sheaves. In the case of cotangent bundles, the proof consists of showing that certain formal properties characterize these categories uniquely, and then verifying these formal properties separately on the two sides. We then deduce the general stably polarized case from the case of cotangent bundles using a doubling construction at infinity.
Federica Pasquotto (Leiden University)
Title: Non-compact Rabinowitz-Floer homology
Abstract: Over the last 30 years, Floer-theoretic invariants have been responsible for many important results in symplectic and contact topology. While these invariants have all been originally defined for compact manifolds, when considering applications one is often interested in situations where the compactness assumption fails to be satisfied. This motivates several recent, interrelated efforts to extend the definition and the computation of Floer-type symplectic invariants beyond the compact setting.
This talk focuses on Rabinowitz Floer homology and its definition for a class of non-compact hypersurfaces in standard symplectic space, which have received the name tentacular. I will describe the challenges arising from the lack of compactness and discuss how additional assumptions enable us to obtain the necessary bounds on moduli spaces of Floer trajectories, and thus extend the definition of the homology. The content of the talk is joint work with Rob Vandervorst and Jagna Wiśniewska.
Thomas Rot (VU Amsterdam)
Title: Proper (co)homotopy in infinite dimensions.
Abstract: Cohomotopy sets of manifolds (homotopy classes of maps into spheres) can be studied via the Pontryagin-Thom construction. The Pontryagin-Thom construction relates the cohomotopy sets to framed submanifolds up to framed cobordism in the domain. I will discuss generalizations of this construction to proper homotopy classes between non-(locally!)-compact spaces. In particular I will discuss a classification of proper homotopy classes of Fredholm mappings between Hilbert manifolds and Hilbert space. This is joint work with Alberto Abbondandolo.
Sara Venkatesh (Stanford)
Title: Leaf-wise intersection points in negative line bundles
Abstract: A leaf-wise intersection point is a certain type of fixed point that can occur when one perturbs a dynamical system. Existence results for leaf-wise intersection points have been proved for a variety of dynamical systems on exact symplectic manifolds. In this talk, we explore the question: what can be said about leaf-wise intersection points in monotone symplectic manifolds? Adapting Floer techniques discovered by Albers, we prove an existence result for Hamiltonian flows on some negative line bundles. We finish with a discussion about the relationship between closed Lagrangian submanifolds and leaf-wise intersection points.
Lei Zhao (Augsburg University)
Title: Regularization of Kepler Problem in Extended Phase Space and Applications
Abstract: In this talk, I shall illustrate the necessity of studying Rabinowitz-Floer Homology on non-compact hypersurfaces with a class of concrete examples arise from celestial mechanics: the regularization of Kepler Problem and its perturbations in extended phase space. As applications, I shall review some results concerning the existence and enumeration of periodic orbits (with possible collisions with the center regularized).