"The KAM theorem, following the original approach designed by Kolmogorov" A minicourse on KAM theory, delivered in the framework of the Pisa-Hokkaido-Roma2 school on Mathematics and its applications Contents of the lectures

• Monday 3 September (2018) - handwritten notes about this lecture
  • A short introduction to the Hamiltonian formalism. Poisson brackets. Canonical transformations.
    Ref.: Chap. 1 and Chap. 2, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • A short discussion about the Liouville theorem and its extension provided by Arnold and Jost.
    Ref.: Sections 3.2 and 3.5 of Chap. 3, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • Introduction of the general problem of the dynamics. Poincaré theorem on the non-existence of first integrals (sketch of the proof, discussion of the consequences).
    Ref.: Sections 4.3 and 4.4 of Chap. 4, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  
  
• Wednesday 5 September (2018) - handwritten notes about this lecture
  • Statement of the KAM theorem. Main ideas behind it. Kolmogorov normal form.
    Ref.: Sections 7.1 and 7.2.2 of Chap. 7, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • Basics of diophantine theory.
    Ref.: Section 4.2.3 of Chap. 4, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • Lie series as a useful tool for defining near to identity canonical transformations: a purely formal introduction. Discussion of the exchange theorem (by Gröbner).
    Ref.: Sections 6.1, 6.2.1 and 6.2.2 of Chap. 6, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  
  
• Friday 7 September (2018) - part I - handwritten notes about this lecture
  • Analytical settings that are necessary to study the convergence of Lie series: complexified domains, weighted Fourier norms, estimates for (multiple) Poisson brackets.
    Ref.: Sections 6.4.2 and 6.7 of Chap. 6, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • Formal algorithm constructing the Kolmogorov normal form, by using the convergence method typical of the classical series. Accumulation of the small divisors (short discussion).
    Ref.: Section 7.2.3 of Chap. 7, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  
  
• Friday 7 September (2018) - part II - handwritten notes about this lecture
  • Proof of the KAM theorem, by using the so called quadratic (or Newton-like or super-convergent) method: reformulation of the algorithm constructing the Kolmogorov normal form, iterative lemma, estimates about the smallness of the perturbation.
    Ref.: Section 7.5 of Chap. 7, making part of the notes about "Hamiltonian Systems", written by A. Giorgilli.
  • Relaxing the hypotheses of the KAM theorem. A short and partial overview of the state-of-the-art in the scientific literature.

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