SUMMER SCHOOL
Hamiltonian Mechanics and Integrable Systems

Topics for the participants' lectures

Every participant will lecture on one of the following papers. Some papers are longer and will be discussed by two of the participants. Some of the papers will only be read partially, more detailed instructions are below. The papers are not listed in the order of difficulty nor in the order they will be presented, but somewhat ordered according to the subject. If you have problems obtaining any of these papers, the organizers can provide you with copies.

Every participant will lecture for 2 hours, split into two lectures of one hour each, one in the first half and one in the second half of the conference week.

Every participant submits a 4-6 page summary of his or her topic prior to the summer school. Preferrably in some form of tex or latex. A sample file can be retrieved here. Deadline for submission of the summary is August 30.

1. Deift, Zhou: "A steepest descent method for oscillatory Riemann Hilbert problems. Asymptotics for the mkdv equation." Annals of Math. (2) 137 (1993), no. 2, 295-368. Part I. This paper introduces an important method to do "nonlinear stationary phase" via a steepest descent method for Riemann Hilbert problems. The results of this method applied to mKdV appear in theorem one on page 303. As the paper is very long , we will only read up to (including) section four, which should be enough to get the main idea. Part I will be in charge of Sections 1 and 2 (pages 306-330) Section 0 "Introduction" will be shared between Parts I and II, in the sense each presenter should give a brief introduction to the material most relevant to his/her part.

   [Tonci Crmaric]

2. Deift, Zhou: "A steepest descent method for oscillatory Riemann Hilbert problems. Asymptotics for the mkdv equation." Annals of Math. (2) 137 (1993), no. 2, 295-368.
   Part II. See Part I for a general description. Part II will be in charge of Sections 3 and 4 (pages 330-354) Section 0 "Introduction" from page 295-306, will be shared between Parts I and II, in the sense each presenter should give a brief introduction to the material most relevant to his/her part.

   [Svetlana Roudenko]

3. Garnett, Trubowitz: "Gaps and bands of one-dimensional periodic Schrödinger operators." Comment Math. Helv. 59 (1984), no. 2, 258-312.
   The purpose of this paper is to characterize all possible spectra of periodic Schrödinger operators, in terms of "gaps" and "bands" The purpose is to prove Theorems 1,2,3. Section 6 and following sections should be skipped, since the paper is too long and Section 6 has been simplified in the follow-up paper by the same authors (Comment. Math. Helv. 62 (1987), no. 1, 18-37). This will leave a small gap in Theorem 3, which, only if time allows, could be filled using that simplification.

   [Kaihua Cai]

4. Christ, Colliander, Tao: "Asymptotics, frequency modulations, and low regularity ill-posedness for canonical defocusing equations." Amer. J. Math 125 (2003), 1235-1293. We care about NLS, the presentation should basically prove Theorem 2. So we only need to read up to (including) Section 4. http://front.math.ucdavis.edu/math.AP/0203044

   [Stefanie Petermichl]

5. H. Hofer and E. Zehnder: "A new capacity for symplectic manifolds", Analysis et Cetera, 405-428, Academic Press, Boston, MA, 1990.
   (See also H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkh\"auser, Basel, Boston, Berlin.) The main task is to prove Theorem 3. Then elaborate the following applications to the extend time allows: Gromov non-squeezing, Proposition 4, Theorem 1. Slides

   [Olga Radko]

6. J. Bourgain: "Approximations of solutions of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties." IMRN 1994, no. 2, 79-90. Prove propositions 1 and 2. Gromov non-squeezing is proved in the Hofer-Zehnder paper presented by another participant.

   [Monica Visan]

7. M. J. Ablowitz, S. Chakravarty, and R. G. Halburd: "Integrable systems and reductions of the self-dual Yang-Mills equations." J. Math. Phys. 44 (2003), no. 8, 3147-3173.
   Explain (self-dual) Yang-Mills and give the reductions to KdV, NLS, Euler-Arnold-Manakov, and Painlevé. You should also describe what makes the Painlevé equations special (e.g. see E. L. Ince, Ordinary Differential Equations. Dover Publications, New York, 1944).

   [Brett Wick]

8. V. Arnol'd: "Instability of dynamical systems with several degrees of freedom." Soviet Math. 5 (1964), no.3, 581-585. (Russian original in Doklady Akad. Nauk SSSR vol. 156)
   AND
   H. P. McKean: "Compatible Brackets in Hamiltonian Mechanics", Important developments in soliton theory, 344--354, Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1993.
   These two papers have nothing in common, except that they are both short. The first is the original paper on Arnol'd diffusion, which provides a counterpoint to both KAM theory and Nekhoroshev estimates. The second gives a quick introduction to the bihamiltonian approach to integrable systems. Sections 11-14 may be omitted. Arnold slides McKean slides

   [Irina Nenciu]

9. M. Levi and J. Moser: "A Lagrangian proof of the invariant curve theorem for twist mappings", Smooth ergodic theory and its applications, 733--746, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.
   This paper gives an illustration of KAM techniques. The goal should be to present the entire paper, though Sections 8.A and 8.B may be sketched if necessary; Section 8.C can be omitted. While the paper is self-contained, a little background reading on KAM should help clarify the essential ideas. See J. Pöschel, "A lecture on the classical KAM theorem", in the same proceedings; and C. E. Wayne, "An introduction to KAM theory", Dynamical systems and probabilistic methods in partial differential equations, 3--29, Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, 1996.

   [Silvius Klein]

10. J. Pöschel: "On Nekhoroshev's estimate at an elliptic equilibrium." Internat. Math. Res. Notices 1999, no. 4, 203--215.
    (Also available through the author's web page.)
    Present the entire paper.

    [Eric Ryckman]

11. J. Bourgain: "Construction of Quasi-Periodic Solutions for Hamiltonian Perturbations of Linear Equations and Applications to Nonlinear PDE" International Math Research Notices 1994 No. 11 pp. 475-497 We like to understand Theorems 1 and 2 (NLS). Section 5 on the wave equation may be skipped.

    [Dmitry Pavlov]