Abstracts
Oliver Fabert: Generalizing symplectic topology from 1 to 2 dimensions
Floer theory is used to prove rigidity results about solutions of the Hamilton equations on symplectic manifolds, whose basic examples are geodesics on Riemannian manifolds. Minimal surfaces are the two-dimensional generalizations of geodesics. In my talk I will introduce the corresponding generalizations of symplectic manifolds and real-valued Hamiltonian systems which allow for a generalisation of Hamiltonian Floer theory. Further I will discuss why the standard polysymplectic approach needs to be modified for Floer theory to exist and I will illustrate how holomorphic Hamiltonian systems and the holomorphic Fukaya category of Kontsevich-Soibelman are actually part of the modified geometric framework.
Urs Frauenfelder: Doubly symmetric orbits are good
This is joint work with Agustin Moreno. Doubly symmetric periodic orbits play an important role in many problems in celestial mechanics as well as in the dynamics of atoms. For example the direct and retrograde periodic orbit in Hill's lunar problem are doubly symmetric or Langmuir's periodic orbit in the dynamics of the Helium atom is doubly symmetric. In the talk we show that a doubly symmetric periodic orbit is never negative hyperbolic so that all its iterates are good orbits in the sense of Symplectic Field Theory (SFT). The proof of this result uses a real version of Krein theory and the GIT quotient.
Mar Giralt Miron: Chaotic dynamics to L3 in the Restricted 3-Body Problem
In the Restricted Planar Circular 3-Body Problem, which models the dynamics of a massless body influenced by two massive bodies in circular orbits, the Lagrange point L3 is a saddle-center critical point. We explore the family of periodic orbits (close enough) to L3 and prove that these orbits intersect transversally, leading to chaotic dynamics. Furthermore, we identify a generic unfolding of a quadratic homoclinic tangency, which gives rise to Newhouse domains. The talk is based on a joint work with Inma Baldomá and Marcel Guardia.
Daniel Jaud: Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries
We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have a focus at the Kepler center, or have its center at the Hookian center respectively. When the particle hits the boundary it is ideally reflected according to the law of reflection. These systems are known to be integrable.
We describe the consecutive billiard orbits in terms of their foci. We show that the second foci of these orbits always lie on a circle in the Kepler case. In the Hooke case, we show that the foci of the orbits lie on a Cassini oval. For both systems we analyze the envelope of the directrices of the orbits as well.
Holger Waalkens: The charged three-body problem
The charged three-body problem is a generalization of the Newtonian three-body problem where the coefficients in the potential are not necessarily given by products of masses but arbitrary real numbers that in addition to gravitational interactions can also describe, e.g., Coulomb interactions. In this talk we discuss the existence of relative equilibria in the charged three-body problem, their relation to central configurations and their manifestations in the bifurcations of Hill regions. The talk is based on joint work with Igor Hoveijn.
Yuchen Wang: Dynamics of concentrated vortex patch in planar bounded domain
Vortex patch is a special class of weak solutions for the two-dimensional incompressible Euler equations. This is an infinitely dimensional Hamiltonian system with non-canonical symplectic structure. Because of its strong nonlinearity, dynamics of vortex patch has not been fully understood yet.
In this talk, I shall introduce our studies on the local dynamics of concentrated vortex patches in bounded domains, including existence of steady solutions and their stability. The is mainly based on perturbation arguments.
Xingbo Xu: Determination of the doubly symmetric periodic orbits
Doubly-symmetric families of periodic orbits exist In the circular restricted three-body problem as well as in The Hill’s lunar three-body problem. We talk about the Periodicity conditions and do some numerical computation Of some comet-type and Hill-type periodic orbits. We also study some periodic orbits near the equilibria points. The main talk is based on the work Xu(2023, Celes Mech).
Zhengyi Zhou: Kahler compactification of C^n and Reeb dynamics
We will present two results in complex geometry: (1) A Kahler compactification of C^n with a smooth divisor complement must be P^n, which confirms a conjecture of Brenton and Morrow (1978) under the Kahler assumption; (2) Any complete asymptotically conical Calabi-Yau metric on C^3 with a smooth link must be flat, confirming a folklore conjecture regarding the recognition of the flat metric among Calabi-Yau metrics in dimension 3. Both proofs rely on relating the minimal discrepancy number of a Fano cone singularity to its Reeb dynamics of the conic contact form. This is a joint work with Chi Li.