Jeudi 16 novembre 2023
Salle de conférences IRMA
Le 16 novembre 2023 aura lieu à l'IRMA une “journée symplectique” conjointe Strasbourg-Heidelberg.
Organisateurs : Peter Albers, Mihai Damian, Agustin Moreno, Alexandru Oancea, Emmanuel Opshtein, Margherita Sandon.
Lieu : salle de conférences de l’IRMA
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Jeudi 16 novembre 2023
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09:00
Tobias Witt, Heidelberg
Global surfaces of section for non-convex billiards
Following an approach by Benci-Giannoni and Albers-Mazzucchelli, Euclidean billiards can be approximated by the motion of point particles in a sequence of smooth but increasingly steep potentials, with the build-up of a repulsive potential wall used to model the boundary of the billiard table. By establishing a precise relation between the Morse and Conley-Zehnder indices of (possibly degenerate) orbits, we argue that for certain billiard tables the Reeb flow of the approximating smooth system will become dynamically convex. As a corollary we recover a classical result by Benci-Giannoni that every 2-dimensional billiard table (with smooth boundary) admits a periodic orbit with at most 3 reflections. -
09:30
Nicolas Stutz, Strasbourg
The Engel geometry of the space of pointed lightlike geodesics of a spacetime
The aim of this talk is to explore some of the geometric structures that can be carried by the space of lightlike geodesics of a Lorentzian manifold of dimension 3. We will start by defining these objects and quickly explain how a contact structure appears on this space. We will then turn to the space of pointed lightlike geodesics and see that this space is provided with an Engel structure, constructed as a prolongation of the previous contact structure, and we will describe the resulting Engel structure geometrically. We will base our talk on work by A. Marìn-Salvador and R. Rubio (Preprint arXiv:2112.06955, (2021)). -
10:00
Arthur Limoge, Heidelberg
Floer homology and the Three-body problem
We discuss the development of Floer-theoretical tools in the study of the three-body problem. In particular, we aim to define a local version of wrapped Floer homology; in a mission to search for collision orbits. -
10:30
Pause café -
11:00
Robin Riegel, Strasbourg
Homology with A-infinity coefficients and Chas-Sullivan product
In this talk, I will briefly describe the construction introduced by J-F.Barraud, M.Damian, V.Humilère and A.Oancea in the article "Morse Homology with DG coefficients" showing that one can use the homological data of higher dimensional moduli spaces of Morse trajectories in chains in order to define a twisted homology theory with differential graded coefficients equipped with a right module structure on the cubical complex of based loops. I will then explain how to extend this construction to coefficients in an A-infinity module structure on the complex of based loops. Finally, I will focus on an application of this setting, which consists in giving a chain level description of the Chas-Sullivan product on the cubical complex of the free loop space. -
11:30
Levin Maier, Heidelberg
On Mañé's critical value for the Hunter-Saxton system
We will study magnetic deformations of the Hunter-Saxton system, in the sense of magnetic geodesic flows. We represent this system as a Hamiltonian flow on an infinite dimensional Lie group and use this to study blow ups and construct global weak solutions of this system of nonlinear partial differential equations. Furthermore we will use the global weak magnetic flow on the infinite dimensional Lie group to prove that any two points in there can be connected by a magnetic geodesic as long as the strength of the magnetic field is less then Mañé's critical value. -
12:00
Colin Fourel, Strasbourg
Degeneration of the Leray-Serre spectral sequence for Hamiltonian fibrations
In 1968, Deligne proved that the Leray-Serre spectral sequence with rational coefficients degenerates for a smooth projective morphism between complex algebraic varieties, using sheaf cohomology and the hard Lefschetz theorem. In 1999, Lalonde, McDuff and Polterovitch proved degeneration for Hamiltonian fibrations over the 2-sphere, using moduli spaces of holomorphic sections. Very recently, Abouzaid-McLean-Smith generalized this to integer coefficients, and more generally to any complex-oriented extraordinary cohomology theory. We will explain and compare the two different approaches, in both the algebraic and symplectic setting. -
12:30
Déjeuner -
14:30
Agustin Moreno, Heidelberg
Symplectic methods in space mission design
In this talk, I will illustrate how methods from symplectic geometry can be leveraged in the early stages of space mission design, i.e. when one is trying to choose a periodic trajectory in which to place a spacecraft in orbit. I will discuss the same “symplectic toolkit” that I have presented to the engineers at NASA responsible for missions to Europa and Enceladus. Based on joint work with Cengiz Aydin (Heidelberg), Urs Frauenfelder (Augsburg), Dayung Koh (JPL, NASA), Otto van Koert (Seoul). -
15:30
Pause café -
16:00
Emmanuel Opshtein, Strasbourg
Liouville polarizations and their Lagrangian skeleta in dimension 4
In the simplest framework of a symplectic manifold with rational symplectic class, a symplectic polarization is a smooth symplectic hypersurface Poincaré-Dual to a multiple of the symplectic class. This notion was introduced by Biran, together with the isotropic skeleta associated to a polarization, and he exhibited symplectic rigidity properties of these skeleta. In later work, I generalized the notion of symplectic polarizations to any closed symplectic manifold, and showed that they are useful to construct symplectic embeddings. In the present talk, I will explain how this notion of polarization can be generalized further to the affine setting in dimension 4 and how it leads to more interesting embedding results. These refined embedding constructions provide a new way to understand the symplectic rigidities of Lagrangian skeleta noticed by Biran and get new ones. These results also lead to (seemingly new kind of) rigidities for some Legendrian submanifolds in contact geometry. I will present several examples and applications. Work in progress, in collaboration with Felix Schlenk.