Program

There will be an 8-hours introductory lecture about differential and Riemannian geometry by M. Crampon.
Then there will be four more specialized courses, by P. Herreros, A. Sambarino, S. Sandon and F. Schlenk, each 6 hours.
Each specialized course will have two 1-hour sessions of exercises.

Here is a tentative schedule:

	Thursday 25.9.	Friday 26.9.	Saturday 27.9.	Sunday 28.9.	Monday 29.9.	Tuesday 30.9.	Wednesday 1.10.	Thursday 2.10.	Friday 3.10.
9:30-10:30	M. Crampon	M. Crampon	M. Crampon	free	A. Sambarino	P. Herreros	A. Sambarino	S. Sandon	S. Sandon
10:30-10:50	coffee	coffee	coffee		coffee	coffee	coffee	coffee	coffee
10:50-11:50	M. Crampon	M. Crampon	M. Crampon		A. Sambarino	P. Herreros	A. Sambarino	S. Sandon	S. Sandon
12-12:45	Exercices	Exercices	Exercices		Exercices	A. Sambarino	S. Sandon	Exercices	Exercises
12:45-14:30	lunch	lunch	12:45-18:		lunch	lunch	lunch	lunch	lunch
14:30-15:30	M. Crampon	P. Herreros	trip to		P. Herreros	Exercices	S. Sandon	F. Schlenk	F. Schlenk
15:30-15:50	coffee	coffee	"San Cristobal hill"		coffee	coffee	coffee	coffee	coffee
15:50-16:50	Exercises	P. Herreros	with "empanadas"		P. Herreros	A. Sambarino	Exercices	F. Schlenk	F. Schlenk
17-17:45	Exercices	Exercices	on the top		Exercices	Exercices	F. Schlenk	Exercices	Exercices

All lectures will take place at the auditorium of the Mathematics building at the University of Santiago de Chile. See "Practical Infos" for how to get there.

Abstracts

Mickaël Crampon : Introduction to Differential and Riemannian Geometry

In this preparatory lecture, we want to introduce the main objects that are needed for the more specialized lectures. In differential geometry, this includes the notions of differentiable manifold, tangent and cotangent bundles, vector fields... On the Riemannian side, we will define what is a Riemannian metric and will then focus on the more basic objects such as covariant derivative and geodesics. We will also present the Hamiltionian point of view on Riemannian geometry, that will allow us to reveal the symplectic nature of the cotangent bundle of a manifold. Curvature will be presented from a non-tensorial point of view, and we will insist on the most useful consequences. Cours

Pilar Herreros : Boundary Rigidity Problems

Boundary rigidity, in general, refers to the question of whether the metric on a manifold is determined by some data on the boundary. The most commonly studied is distance boundary rigidity for simple domains, where the boundary data known is the distance between any two points in the boundary. In cases where the distance is not enough to determine the metric, we can study instead scattering rigidity where the known boundary data is the scattering data of the region; i.e. for each point and inward direction on the boundary, it associates the exit point and direction of the corresponding unit speed geodesic. We will focus on examples and some cases where rigidity is known and others where it fails, and understanding the difficulties and obstructions.

Andrés Sambarino : Asymptotic properties of negative curvature

In these lectures we will address the following questions concerning geodesics on a compact manifold M. Fix some number T>0: 1. How does the number of closed geodesics of lenght smaller than T grow when T goes to infinity? 2. Given two points x,y in M, how does the number of geodesics from x to y of lenght smaller than T grows when T goes to infinity? Margulis gives precise answers to these questions when the manifold is negatively curved, via deep relations between the dynamics of the geodesic flow and the geometry of the manifold itself. The purpose of the lectures is to give a complete ("self contained") answer for surfaces of constant curvature -1. Ejercicios

Sheila Sandon : Generating functions and the Arnold conjecture for fixed points of Hamiltonian diffeomorphisms

Let M be a compact symplectic manifold. The Arnold conjecture states that any Hamiltonian diffeomorphism of M should have at least as many fixed points as the minimal number of critical points of a smooth function on M. This conjecture was posed in the 60's, and since then has been a great source of motivation for the development of modern symplectic topology. Following the work of Chaperon, Laudenbach, Sikorav and Théret, in my course I will present a proof of the Arnold conjecture in the cases of CP^n and T^2n. The proof will be based on the technique of generating functions, which is a simple but deep technique that relates properties of Hamiltonian diffeomorphisms to the Morse theory of some associated functions. I will also discuss a version of the Arnold conjecture for Lagrangian intersections in the cotangent bundle and, if I will have time, some other related topics such as the Conley conjecture for periodic points of Hamiltonian diffeomorphisms of T^2n (following Théret and Mazzucchelli) and Gromov's Non-squeezing Theorem in R^2n (following Viterbo). The course does not require any previous knowledge of symplectic topology. Notes

Felix Schlenk : Closed orbits of classical mechanical systems via the minimax principle

Closed geodesic on a closed Riemannian manifold M can be easily found in each conjugacy class of the fundament group of M. However, if M is simply connected, then finding closed geodesic is much harder. Birkhoff had the beautiful idea to apply a minimax technique (the mountain pass lemma) to this problem (in the case of spheres). Closed geodesics are the orbits of a free particle. If on looks, more generally, for closed orbits of a particle subject to a potential force and a magnetic field, then (in contrast to the geodesic situation) the dynamics depends on the energy of the particle. Minimax techniques can still be used to prove existence of closed orbits on almost every energy level. The beauty of this theory is that, for different energy regimes, different minimax classes must be chosen. Notes

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