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Topics on Nash-Moser and KAM theory

Course Type: 
PhD Course
Academic Year: 
June 2021
20 h

The course deals with Hormander's approach to implicit function theorems of Nash-Moser type and to its recent extensions. We will consider scales of function spaces of both Holder and Sobolev classes, and their two different mechanisms leading to sharp regularity solutions. Strong relations with the Littlewood-Paley decomposition will be emphasized, as well as the role of orthogonality in the L^2-based case. We will discuss the algebraic part of the scheme, also in comparison with the (more standard) Moser's approach. We will make a comparison between scales parametrized by regularity and scales where the parameter is a summability power, and we will present a simple, recent non-existence result for families of smoothing operators in the ell^p scale. In the last part of the course we will discuss some applications of the Nash-Moser-Hormander method, and we will see how it can be considered as a general approach to Cauchy problems and control theory for evolution PDEs, alternative to the para-differential approach. Time permitting, I will show some application of the Nash-Moser-Hormander method to some singular perturbations problems, and/or something from a work in progress on the existence of KAM tori for finite-dimensional nearly integrable Hamiltonian systems at the threshold regularity. 

Room 133 (Room 005 for the days 08 and 15 June, Room 134 for the days 10 and 17 June)
Next Lectures: 

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