Titles and Abstracts of short courses in the workshop.
1. Professor Alain Albouy (Astronomie et Systemes Dynamiques, IMCCE, Paris)
TITLE: Central configurations-Attempts to solve a wonderful system of equations.
ABSTRACT: CENTRAL CONFIGURATIONS are the possible configurations in a state of RELATIVE EQUILIBRIUM in the classical n-body problem. A relative equilibrium is a motion where the mutual distances between the particles remain constant. Such is the famous equilateral triangle of Lagrange, made of three particles with arbitrary mass in the plane, in a uniform rotation around their center of mass.
A remarkable fact is the existence of a system of equations that defines the configuration independently of the velocities and the dimension of the space of motion. In principle this system
has as many equations as unknowns, and the configurations (defined up to isometry, for example by the data of the mutual distances) are isolated. However, the possibility of a continuum of central configurations has never been excluded. This question goes back to Chazy and Wintner, and has been chosen by Smale as his 6th problem for the 21st century. The main questions are open already with four masses in the plane.
I will try to present different ways of writing the equations, and to extract in each case many informations on the solutions. The mutual distances form an interesting set of unknowns. Dziobek introduced some associated variables, the areas of the subtriangles, in the case of 4 particles in the plane. The algebraic structure of the system in these variables is quite interesting: one can apply the theory of the Legendre transform, and the structure of the space of mutual distances, as described by Chenciner and myself, adds some new aspects to this theory.
We will also describe some results of symmetry, in different cases, starting with a detailed study of the collinear 3-body case. We will also discuss the techniques of computer algebra (resultant and Sturm algorithm), as well as the possibilities offered by the "fewnomials" and Bernstein-Kushnirenko-Khovanskii theory.
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2. Professor Eric Sere (Universite Paris-Dauphine)
TITLE: Stationary solutions of nonlinear Dirac systems
ABSTRACT: The Dirac equation plays a fundamental role in relativistic quantum mechanics. In this short course, we consider nonlinear models of the electron based on this equation, and we look for stationary solutions. These solutions are found as critical points of an energy functional,
which presents strong similarities with the action functional of Hamiltonian Mechanics.
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3. Professor Jiangong You (Nanjing University)
TITLE: An introduction to stability theory of nearly integrable Hamiltonian systems
ABSTRACT: In 8 – 10 hour lectures, I am going to give a brief introduction to the stability theory of nearly integrable Hamiltonian systems, including
(1) some basic concepts in Hamiltonian dynamics, integrable systems;
(2) normal form and average method;
(3) exponential stability (Nekhoroshev estimates);
(4) KAM theory.
References:
[1] V.I.Arnold, Geometrical methods in the theory of ordinary differential equations.
[2] A. Georgilli, Notes on exponential stability of Hamiltonian systems, homepage of A. Georgilli: www.matapp.unimib.it/~antonio
[3] J. Poschel, A lecture on the classical KAM-theorem, homepage of J.Poschel: www.poschel.de
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4. Professor Chongchun Zeng (University of Verginia)
TITLE: Hamiltonian PDEs and perturbation problems
ABSTRACT: In these lectures, we will introduce some basic geometric Hamiltonian PDEs, such as the wave map and Schrodinger map equations. These are basically vector valued wave and Schrodinger equations, where the ranges of the unknown functions are Riemannian or Kahler manifolds. The main part of the lectures will be on perturbation problems, particular singular perturbations given by strong constraining potentials. These strong potentials come naturally from physics. They constrain the motions in neighborhoods of some submanifolds and create multiple scales of motions: regular waves and fast oscillations. We will discuss the basic convergence to the singluar limits and also the dynamics.
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5. Professors Yiming Long, Chungen Liu, and Chaofeng Zhu (Nankai University)
TITLE: An introduction to an index theory of symplectic paths with applications.
ABSTRACT: This mini-course contains a brief introduction to the index theory of symplectic paths and its applications to nonlinear Hamiltonian systems. The main topics in this course include
(1) Basic concepts on symplectic matrix, groups, and variational methods for Hamiltonian systems.
(2) The index theory for symplectic paths.
(3) Iteration theory of the index theory
(4) Applications of the index theory to nonlinear Hamiltonian systems.
The aim of these lectures is to give students an intuitive idea what the index and its iteration theory are, and how to use them to solve certain nonlinear problems in Hamiltonian systems and related fields.
REFERENCE: Yiming Long: Index Theory for Symplectic Paths with Applications. Progress in
Math. Vol 207, Birkhauser, Basel, 2002.
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6. Professor Junping Shi (College of William and Mary)
TITLE: Asymptotic Spatial Patterns and Entire Solutions of Semilinear Elliptic Equations
ABSTRACT: In singularly perturbed reaction-diffusion equations on bounded domains, when the diffusion coefficient tends to zero, the behavior of the steady state solutions depends on the qualitative properties of solutions of elliptic equations on the whole space or the half space. The bounded solutions of $\Delta u+f(u)=0$ on the whole space or the half space determine the local asymptotic spatial behavior of solutions to singularly perturbed problems. We will survey results on entire solutions from the classical Liouville theorem, radially symmetric solutions, to recent development on De Giorgi's conjecture. In these earlier works, the nonexistence of patterns has shown for certain nonlinearities, and the typical patterns found are either radially symmetric or monotone. In the talk, periodic patterns and saddle solutions from the speaker's work will be introduced in details, and related conjectures will also be discussed at the end.
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7. Professor Weiping Li (Oklahoma State University)
TITLE: Product structure in symplectic Floer homology
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