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Room P3.10, Mathematics Building
Rafael de la Llave, Georgia Tech.
Manifolds on the verge of a regularity breakdown
There are two main stabibility 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 transition and "renormalization group" provides a
unifying point of view. Nevertheless, many questions remain open.