Manifolds on the verge of a regularity breakdown
Rafael de la Llave, School of Mathematics of Georgia Tech
21 de diciembre, 2012. Departamento de Matemáticas, Aula 520, 11:30.
There are two main stability arguments for solutions in dynamical systems: the theory of normal hyperbolicity and the Kolmogorov-Arnold-Moser theory of perturbations.
In recent times, there has been progress in developing versions of the theory that are well suited for computations. The theory does not require that the system is close to integrable, but rather uses geometric identities. The theorems prove that approximate solutions satisfying some non-degeneracy assumptions correspond to a true solution. Furthermore, the proofs lead at the same time to very efficient algorithms.
Implementing these algorithms, leads to some conjectural insights on the phenomena that happen at breakdown. They turn out to be remarkably similar to phenomena that were observed in phase transitions and the "renormalization group" provides a unifying point of view. Nevertheless, many questions remain open.