Georgia Institute of Technology College of Sciences

We will present the "a-posteriori" approach to KAM theory.

We formulate an invariance equation and show that an approximate-enough solution which verifies some non-degeneracy conditions leads to a solution.  Note that this does not have any reference to integrable systems and that the non-degeneracy conditions are not global properties of the system, but only properties of the solution. The "automatic reducibility" allows to take advantage of the geometry to develop very efficient Newton methods and show that they converge.

This leads to very efficient numerical  algorithms (which moreover can be proved to lead to correct solutions), to validate formal expansions. From a more theoretical point of view, it can be applied to other geometric contexts (conformally symplectic, presymplectic) and other geometric objects such as whiskered tori. One can deal well with degenerate systems, singular perturbation theory and obtain simple proofs of monogenicity and Whitney regularity.

This is joint work with many people.

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

In this thesis, the Rayleigh-Taylor instability effects in the setting of the Navier-Stokes equations, given some three-dimensional and incompressible fluids, are investigated. The existence and the uniqueness of the temperature variable in the the weak form is established under suitable initial and boundary conditions, and by the contraction mapping principle we investigate further the conditions for the solution to belong to some continuous class; then a positive minimum temperature result can be proved, and with the aid of the RT instability effect in the density and the velocity, the instability for the temperature is established.