Constructive methods in KAM theory- from numerics to regularity

CDSNS Colloquium
Wednesday, May 6, 2020 - 9:00am for 1 hour (actually 50 minutes)
Attendee link:
Rafael de la Llave – Georgia Tech – rafael.delallave@math.gatech.edu
Alex Blumenthal

Please Note: This is the first installment of our CDSNS virtual colloquium, which will be held in a Bluejeans event space on Wednesdays at 9AM (EST).

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.