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Nash-Moser implicit function theorems and KAM theory

Course Type: 
PhD Course
Academic Year: 
2024-2025
Period: 
October - December
Duration: 
50 h
Description: 
The main assumption of the classical implict function theorem in Banach spaces is that the linearized operator has a bounded inverse. This is sufficient for constructing bifurcation theory of periodic solutions of finite dimensional dynamical systems. On the other hand, there are several problems where this assumption is not satisfied, i.e. the linearized operator is unbounded, for example for the search of quasi-periodic solutions. To overcome this challenge, the Nash-Moser theory was developed. 
 
In this course we shall prove a Nash-Moser-Zehnder implicit function theorem for the search of zeros of a nonlinear operator acting on scales of Banach spaces, of either analityic or differentiable functions. As applications we shall prove the Siegel linearization theorem for an analytic map close to an elliptic fixed point and the classical KAM (Kolmogorov–Arnold–Moser) theorem concerning the persistence of quasi-periodic solutions for sufficiently small and smooth perturbations of completely integrable, non-degenerate, finite dimensional Hamiltonian systems.
  
Other applications shall be considered time permitting.
 
Next lectures:
Wednesday 16 October, 11-13, room 133

Thursday 17 October, 14-16, room 133

Tuesday 22 October, 11-13, room 136

Thursday 24 October, 14-16, room 133

Tuesday 5 November, 11-13, room 136

Thursday 7 November, 11-13, room 134

Tuesday 12 November, 11-13, room 136

Thursday 14 November, 14-16, room 133

Tuesday 19 November, 11-13, room 136

Thursday 21 November, 14-16, room 133

Tuesday 26 November, 11-13, room 136

Thursday 28 November, 14-16, room 133

Tuesday 10 December, 11-13, room 136

Thursday 12 December 14-16, room 133

Tuesday 17 December, 11-13, room 136

Thursday 19 December 14-16, room 133

Location: 
A-133
Next Lectures: 
Tuesday, November 12, 2024 - 11:00 to 13:00
Thursday, November 14, 2024 - 14:00 to 16:00
Tuesday, November 19, 2024 - 11:00 to 13:00
Thursday, November 21, 2024 - 14:00 to 16:00
Tuesday, November 26, 2024 - 11:00 to 13:00
Thursday, November 28, 2024 - 14:00 to 16:00
Tuesday, December 10, 2024 - 11:00 to 13:00
Thursday, December 12, 2024 - 14:00 to 16:00
Tuesday, December 17, 2024 - 11:00 to 13:00
Thursday, December 19, 2024 - 14:00 to 16:00

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