Georgia Institute of Technology College of Sciences

We consider a class of singular ordinary differential equations, describing systems subject to a quasi-periodic forcing term and in the presence of large dissipation, and study the existence of quasi-periodic solutions with the same frequency vector as the forcing term. Let A be the inverse of the dissipation coefficient. More or less strong non-resonance conditions on the frequency assure different regularity in the dependence on the parameter A: by requiring a non-degeneracy condition on the forcing term, smoothness and analyticity, and even Borel-summability, follow if suitable Diophantine conditions are assumed, while, without assuming any condition, in general no more than a continuous dependence on A is obtained. We investigate the possibility of weakening the non-degeneracy condition and still obtaining a solution for arbitrary frequencies.

TBA

We introduce a quantum version of the Kac Master equation,and we explain issues like equilibria, propagation of chaos and the corresponding quantum Boltzmann equation. This is joint work with Eric Carlen and Maria Carvalho.

We will talk about discrete versions of the Bethe-Sommerfeld conjecture. Namely, we study the spectra of multi-dimensional periodic Schrödinger operators on various discrete lattices with sufficiently small potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we characterize those energies at which gaps may open, and we give sharp arithmetic criteria on the periods that ensure no gaps open. We will also provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. This talk is based on a joint work with Svetlana Jitomirskaya and another work with Jake Fillman.