Please note: the room has changed since the first announcement of this seminar. It will now take place in the Salle Solvay.
Oliver Fabert (Vrije Universiteit Amsterdam) will speak in the geometry seminar on January 24th, at 13h30, in the Salle Solvay (5th floor building NO). His title is “The Arnold conjecture in infinite dimensions” and his abstract is below.
For complex projective spaces a version of the Arnold conjecture states that the number of fixed points of every Hamiltonian flow with Hofer norm less than one is greater than the complex dimension. In my talk I will discuss how this result generalizes to the projectivization of the Hilbert space of complex-valued square-integrable functions on the circle. While there exist smooth time-periodic Hamiltonians with Hofer norm less than one without any fixed points, I will show why the corresponding nonlinear Schrödinger equation (obtained by adding the Laplacian term) indeed has infinitely many time-periodic solutions. In order to prove the existence of the relevant holomorphic curves from Floer theory in infinite dimensions, I employ the following bizarre fact from nonstandard model theory: Every infinite-dimensional symplectic Hilbert space is contained in a symplectic vector space which behaves as if it were finite-dimensional.