Georgia Institute of Technology College of Sciences

The purpose of this talk is to discuss our recent work on multi-frequency quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles. Analogous results for one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.

In this talk, using a technique introduced by P.~T.~Nam in 2012 and the Coulomb Uncertainty Principle, I will present the proof of new bounds on the excess charge for non relativistic  atomic systems, independent of the particle statistics. These new bounds are the best bounds to date for bosonic systems. This is joint work with Juan Manel González and Trinidad Tubino.

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The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.

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This presentation, which is based on the work Sun [2], is dedicated to describing the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton mani- fold, denoted by UN . We construct (generalized) action–angle coordinates which establish a real analytic symplectomorphism from UN onto some open convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by G ́erard–Kappeler [1]. The global well-posedness of the BO equation on UN is given by a polynomial characterization and a spectral char- acterization of the manifold UN . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup. The construction of action–angle coordinates for each UN constitutes a first step towards the soliton resolution conjecture of the BO equation on the line.