Abstract: In the last years, there was a remarkable convergence between three seemingly remote fields of mathematics: theoretical computer science, geometry of Banach spaces, K-theory of C^*-algebras. The common theme is embeddings of discrete metric spaces into Hilbert or Banach spaces. Learning of techniques from other fields allowed for mutual cross-fertilization, and it is the purpose of this set of lectures to present some recent developments in this fascinating subject.

Abstract: The purpose of this course is to dive into the exciting realm of geometric topology.
Classical Teichmueller theory is about hyperbolic structures on surfaces of negative Euler characteristic. As we will see, hyperbolic metrics are characterised by the set of lengths they admit on closed curves. This naturally leads to Thurston's compactification of Teichmueller space, whose new objects are measured foliations. Using these, we will derive Thurston's classification of surface mapping classes into three types: periodic, reducible and pseudo-Anosov.
In practice, measured foliations are most efficiently described via train tracks. We will spend sometimes on constructing these, following a recent approach by Gerber.
On our way, we are going to discover various classical theorems, e.g. Hurwitz' 84(g-1) Theorem, Nielsen's realisation problem, and the theory of fibred knots and links.

Abstract: The goal of the course is to construct symplectic and Lagrangian Floer homology, and to show how these two homologies can be used to solve problems in symplectic topology and Hamiltonian dynamics.
This is a sequel of the course given last year. I will therefore assume that the audience is familiar with Morse theory and Morse homology. (Though, these topics will be sketched again in one or two lectures). I will then give the geometric and topological parts of the construction of symplectic and Lagrangian Floer homology, with some, but not all, analytical details.
In the second half of the course these theories will be applied to prove several existence results for closed orbits in Hamiltonian systems (Arnold and Weinstein conjectures), to give lower bounds on the topological entropy of classical Hamiltonian systems, and to give some results on Lagrangian intersections.