“ The Master in the art of living makes little distinction between his work and his play, his labor and his leisure, his mind and his body, his education and his recreation, his love and his religion. He hardly knows which is which. He simply pursues his vision of excellence in whatever he does, leaving others to decide whether he is working or playing. To him he is always doing both. ”
by Zen Buddhist Text

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Integrable

I have worked for many years on topological/geometric properties of integrable dynamical systems, and also obstructions to integrability. Below are the main results that I’ve obtained:

  • Structure of singularities of integrable Hamiltonian systems
    • Nondegenerate case: topological decomposition theorem, and existence of partial action-angle variables
    • Degenerate case: existence of  torus actions (or partial action variables)
    • Local situation: convergence of Poincaré-Birkhoff normalization for any analytic integrable system (Hamiltonian or non-Hamiltonian)
    • Near a nondegenerate singular orbit: semi-local smooth linearization theorem (joint work with Eva Miranda)
    • Monodromy formula for focus-focus singumarities
  • Characteristic classes & global classification
    • Definition of the Chern class and the Lagrange class (for integrable Hamiltonian systems with singularities). Classification theorem. Surgery construction.
  • Obstructions to integrability
    • Galoisian obstruction to meromorphic non-Hamiltonian integrability (extension of Morales-Ramis-Simo theorem to the non-Hamiltonian case, joint work with Michael Ayoul)
  • Integrable infinite-dimensional systems
    • Birkhoff normal form & topological structure of the focusing cubic integrable nonlinear Schroedinger equation (joint work with T. Kappeler, T. Lohrmann and P. Topalov)
  • Gelfand-Cetlin system
    • Singularities of G-C system: all degenerate singular fibers are smooth homogeneous spaces, and in some case there is also a “canonical model” for the system near these singularities.

Some research problems (for PhD students)

  • Topological structure of singularities of non-Hamiltonian (say non-holonomic) integrable systems. To what extent the results in Hamiltonian setting still hold in this case ? (Currently I have 1 student working on this problem)
  • Random perturbations of integrable systems  (starting with the simplest example: random harmonic oscillators).  Looking for diffusion results for such systems. (Another PhD student started to look at this problem)

This version: 16/Sep/2010

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