For the solution to one-dimensional lattice Schrödinger equation, we prove that the diffusion norm grows linearly with the time, if the potential is periodic or sufficiently small quasi-periodic(with Diophantine frequencies).

We present normal forms for unfoldings of regular nilpotent contact points of slow-fast systems in the plane. The normal forms are useful in the treatment of non-generic jump points and non-generic turning points. For non-generic jump points, we prove a normal form of Lienard type, while for non-generic turning points, the normal form is of Lienard type up to exponentially small error. The techniques being used are based on Gevrey estimates on formal power series, summation up to exponentially small error and Gevrey version of the preparation theorem for right equivalences.

Let M be a real (CR) submanifold of C^n. The traces of holomorphic functions on M turn out to satisfy a certain system of PDEs (the tangential Cauchy-Riemann equations). The solutions of this system are called CR functions: in general they do not reduce to the restriction of holomorphic functions and need not be analytic.In joint work with Bernhard Lamel, we consider some aspects of the behavior of germs of smooth CR functions which exhibit substantial differences from the holomorphic case. First we look at the Borel map, sending a smooth CR function to its formal expansion at 0, and show that in certain cases it is surjective. Furthermore we determine the "critical" vanishing rates of smooth CR functions at a point.