Open problems

Please let me know if you are aware of partial or complete resolutions of these.

  • Intersection between control theory and game theory. Consider the two dimensional heat equation with a forcing term on an open region. Assume the forcing term is the sum of two contributions, each chosen by one player. What is the dynamic when:
    • The player contributions have disjoint support in space, disjoint support in time, both in time and space.
    • One player tries to reach the zero state, the other to prevent it.
    • Both player get assigned a target heat distribution, and try to reach it.
  • Connection between SDE and PDE. The Feynman-Kac formula links Brownian processes to heat equation. Can this link be extended to other PDEs? For example, does there exists a random process whose expectation solves the linear Korteweg-de Vries equation? Personal guess: look for a process with non-zero cubic variation.
  • Spectral measure for differential operator with non-constant coefficients. Heteroclinic waves are associated to a two dimensional continuous spectrum. Can a spectral measure be associated to it? Explicit computation is known for constant coefficients using Fourier transform.

Tools

  • Text comparison software Meld.
  • Time-schedule LaTeX package.
  • Regular expressions tutorial for search and replace.

Internship reports

  • Asymptotic stability for the extended FKPP equation - Report (french)
    Toulouse, France (summer 2019)
  • Nonlinear stability of a periodic wave using phase modulation - Report (french)
    Rennes, France (fall 2018)
  • Burger’s equation - Report (english - spanish)
    Sevilla, Spain (spring 2017)
  • Homotopy and Galois covering - Report (french)
    Leuven, Belgium (spring 2016)