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Abstract: We describe the main directions of research in Hamiltonian dynamics and present a selections of open problems

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Abstract: We explore the role of the hyperbolic and parabolic higher rank trick in the local rigidity theory of abelian affine actions on the torus.

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Abstract: We justify the averaging principle for slow fast systems in the case where the frozen
motion is ergodic on almost every level set. We then discuss the rate of convergence in
the important case where the fast motion is quasiperiodic

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Abstract: We consider orbits in the planar circular restricted 3-body problem containing repeated near-collisions with the smaller primary. A theorem of Font, Nunes, and Simó states that there exists such an orbit shadowing any sequence of collision orbits satisfying certain conditions. This is proven by estimating the Poincaré map from one near-collision to the next and showing that it is a horseshoe map. It follows that there is an invariant hyperbolic set on which this map is conjugated to a subshift on an alphabet of collision orbits.

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Abstract: Motivated by the question of M. Kac: Can you hear the shape of a drum?
We introduce the Laplace spectrum and the length spectrum for convex planar domains. After showing a connection between the two spectra we shall discuss
problems of spectral rigidity, namely, if it is possible to smoothly deform a domain
without changing its spectra.

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Abstract: In a work in progress with Artur Avila and Mikhail Lyubich, we
show that there are maps in the complex Hénon family with a stable
homoclinic tangency. We will first review Newhouse’s phenomenon for
real surface dynamics and related results. Then we explain how this
phenomenon is related to stable intersections between Cantor sets. Then
we will explain a new mechanism on the stable intersections between two
dynamical Cantor sets generated by two classes of conformal IFSs on the
complex plane.