Inauguration of the Laboratory of Geometry and Dynamical Systems

10 May 2017

Sala de Graus, EPSEB, UPC

12.30 Francisco Presas (ICMAT-CSIC)
           Symplectic topology and Hamiltonian dynamics: a love story.

 Historically, symplectic geometry appeared as a geometric setup for Hamiltonian dynamics. The Hamiltonian side influenced the geometric aspects at the beginning of the story. In the eighties, however, the discovery of rigidity phenomena in Symplectic Geometry turned around the table. We want to describe in this talk the conjectures, nurtured by Symplectic rigidity, that have grown up in the last thirty years. Most of them have to do with the existence/non-existence of periodic orbits in general Hamiltonian systems. Finally, we would address some more recent results addressing the existence of minimal flows in Hamiltonian dynamics.

13:30 - 15:00 Welcome Speeches and Cava

16:00: Jacques Féjoz (Université de Paris Dauphine-Observatoire de Paris) 
[This talk is joint with the Seminari de Sistemes Dinàmics UB-UPC]

              On Linear Point Billiards

Motivated by the high-energy limit of the N-body problem we construct a non-deterministic billiard process, whose table is the complement of afinite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed, piecewise linear curve with vertices on the subspaces and changes of directions upon hitting a subspace constrained by `conservation of momentum' (mirror reflection). The itinerary of a trajectory is the sequence of subspaces it hits. We will describe a fundamental theorem of Burago-Ferleger-Kononenko on the finiteness of itineraries, and a description of the space of trajectories having a fixed itinerary, which has emerged recently in a joint work with Andreas Knauf and Richard Montgomery, as a Lagrangian relation on the space of lines in the Euclidean space. Our method combines those of BFK in non-smooth metric geometry, with generating families.

17:30: Closing