Advanced Courses 2011/12

Specific courses which are held in the framework of a master program are also qualified for PhD students. Such courses can be announced on this page using the usual form and are thereby open for all PhD students participating in the Doctoral Program. In order to minimize travelling, it is recommended to organize such courses in a bi-weekly rhythm or as block courses.

Link to the advanced courses of the academic year 2006/2007, 2007/2008, 2008/2009, 2009/2010, 2010/2011.

Graduate Course of the III Cycle: Metric embeddings in Hilbert and Banach spaces
Prof. Alain Valette (Neuchâtel)
Fall term 2011, Wednesdays, 11:15 - 12:30, beginning: September 21
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.

Table of contents:
1. Motivation I: computer science (from the Sparsest Cut problem to the Goemans-Linial conjecture, to the discrete Heisenberg group)
2. Motivation II: topology (from embeddings into Hilbert spaces to the Novikov conjecture, after G. Yu)
3. Distortion for embeddings of finite metric spaces: Bourgain's upper bound.
4. Metric spaces hard to embed: expander graphs.
5. The group connection: equivariant embeddings and the role of amenable groups.
Place: EPFL, room CM 013

Graduate Course of the III Cycle: Train Tracks, Thurston and Teichmueller Theory
Prof. Sebastian Baader (Bern)
Fall term 2011, Wednesdays, 14:00 - 15:15, beginning: September 21
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.
Place: EPFL, room MA 10

Graduate Course of the III Cycle: Floer homology with applications II
Prof. Felix Schlenck (Neuchâtel)
Fall term 2011, Wednesdays, 15:45 - 17:00, beginning: September 21
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.
Place: EPFL, room MA 10