Program

There will be six courses, three introductory courses and three advanced courses. Each course consists of 4h lectures and 3h exercices
All lectures are in English.

Here is a tentaive schedule (PC = Precourse, C = Course, Ex = Exercices):

added later

Introductory courses

Course 1: The length spectrum of a Riemannian manifold

Otto van Koert

In this introductory course we look at the length spectrum of open domains in the plane (the billiard spectrum) and at the length spectrum of closed Riemannian manifolds (namely the set of lengths of periodic geodesics). I will describe the function CF(L) counting the closed geodesics of length not more than L for flat 2-tori and for hyperbolic surfaces. We will also discuss the Poisson summation formula for flat 2-tori relating CF(L) with the spectrum; this is a baby case of the theorems discussed in the advanced course 2. In the last lecture I will introduce the heat and wave equation on domains and closed manifolds.

Course 2: The Laplace spectrum of a Riemannian manifold

Felix Schlenk

This precourse provides the basics on the spectrum of a compact Riemannian manifold (with or without boundary) needed for the advanced main lectures. For didactical and notational reasons, we often restrict ourselves to 2-dimensional Riemannian manifolds. After introducing the Laplace-Beltrami operator, we prove the basic fact that its spectrum forms a real sequence of positive numbers going to infinity. We then prove the minimax principle (Dirichlet quotients) and illustrate it by proving the Faber-Krahn inequality and showing that the first eigenvalue can be made small on long barbells of area one. It is then time to look at examples: Flat tori and spheres, and also a bit at hyperbolic surfaces. If time permits, we shall also give Milnor's examples of iso-spectral tori that are not isometric.

Course 3: Symplectic geometry

Esmaeel Asadi (lectures) and Amin Mohebbi (exercices)

In these basic lectures we provide backgrounds for the advanced course 3. We first define symplectic manifolds and Hamiltonian flows, and prove Liouville's theorem. We then restrict to cotangent bundles, and show why geodesic flows are very special Hamiltonian flows, and how to model the dynamics of a charged particle in a magnetic field as a Hamiltonian flow. We then define closed characteristics, and as a warm-up to Gromov's non-squeezing theorem prove that one cannot instantaneously squeeze a ball of radius 1 into a thinner cylinder by means of a Hamiltonian deformation. We then discuss the Poincaré return map and the Maslov index. We finally define the action functional on the free loop space and show that its critical points are precisely the periodic orbits of the Hamiltonian flow.

Advanced courses

Course 1: Small eigenvalue of negatively curves surfaces

Asma Hassannezhad

The relation between the small eigenvalues and length of simple closed geodesics in negatively curved surfaces is one of the beautiful and important results in spectral geometry. A fundamental result by Schoen, Wolpert and Yau shows that the length of small closed geodesics on a compact negatively curved surface controls the size of small eigenvalues. Their result later was extended to the non-compact case. In 1990, Burger studied the asymptotics of small eigenvalues in terms of the length of simple closed geodesics. He related small eigenvalues of the surface with the eigenvalues of a weighted graph associated with the set of collapsing closed geodesics. Our aim in this minicourse is to discuss some of the main results in this direction and sketch some ideas of the proofs.

Course 2: From the Laplace spectrum to the length spectrum and back

Urs Frauenfelder

Huber had shown in the sixtieth that for a closed hyperbolic surface the Laplace spectrum determines the length spectrum. This results was generalized in the early 1970 by Colin de Verdière to all negatively curved surfaces and then to all generic Riemannian manifolds by him and Chazarain and Duistermaat-Guillemin. Colin de Verdières's proof is tricky, using the heat kernel and the stationary phase technique. The wave equation technique used by Chazarain and Duistermaat-Guilleminis is more powerful and elegant. When specialized to the circle, these results specialize to the Poisson summation formula. Our goal in this minicourse is to explain the content of these results and some parts of the proofs (the heat kernel, and the trace formulas). Not to get lost we shall mostly focus on surfaces of negative curvature.

Course 3: The role of the action spectrum in symplectic dynamics and geometry

Urs Frauenfelder and Felix Schlenk

The goal of this minicourse is to explain the role of the action spectrum in symplectic geometry in the language of action selectors and its applications. We first sketch the construction of Floer homology in an informal way, and explain why in the closed case Floer homology is isomorphic to singular homology. We then use this isomorphism and the natural filtration of Floer homology by action to define an action selector for each Hamiltonian function. The mere existence of an action selector will then readily lead to deep theorems in symplectic geometry and Hamiltonian mechanics: Gromov's non-squeezing theorem, the proof of the Weinstein conjecture in R^{2n} (stating that for almost every compact energy level there is a periodic orbit), almost existence of periodic orbits in magnetic fields, the unboundedness of Hofer's metric on the group of Hamiltonian diffeomorphisms, etc.

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