Geometric and Quantum Topology in Dimension 3

Description

This school is organised as a session of the SMF « Etats de la Recherche ». Its scope is to present the several spectacular advances obtained in 3-dimensional topology over the last few years and to make them accessible to PhD students and researchers in this and in nearby fields.  The school will also be the opportunities to make an assessment of the main conjectures that are still open and the future perspectives. The recent developments in 3-dimensional topology have given rise to different interactions with geometric group theory, symplectic topology, arithmetics and theoretical physics. The school will also present these various interactions that should lead to new collaborations.

State of the art

The study of 3-manifolds and, more specifically of hyperbolic 3-manifolds has made spectacular progress in the last few years. There have been remarkable advances in two different directions: the understanding of their topological and geometric structure, and the introduction of powerful invariants that should detect non homeomorphic manifolds. The most impressive recent results about the geometric and topological structure of 3-manifolds are due to Perelman (proof of the Poincaré conjecture and of Thurston's geometrization conjecture) on one hand and to Agol, Wise, Kahn-Markovic, Bergeron, Haglund, et al. (proof of Thurston's virtual fibration conjecture) on the other. These results are based on analytic methods in the first case (Ricci flow) and on geometric group theory in the second one (CAT(0) cubulations). Interactions with symplectic topology, arithmetics and theoretical physic have led to the definition of different invariants for 3-manifolds (Casson) and 4-manifolds, and for knots (Khovanov and Heegaard-Floer homologies) which are a way to distinguish 3-manifolds. It was for instance shown by Kronheimer and Mrowka in 2010 that Khovanov's homology detects the trivial knot. The Heegaard-Floer homology, too, detects the unknot. Since Manolescu, Ozsvath, Sarkar, Szabo and D. Thurston showed in 2006 that the Heegaard-Floer homology can be defined combinatorically, it has become the first combinatorial invariant to detect the unknot.  

Target audience

The school is addressed mainly to young researchers, PhD students, and postdocs working in low-dimensional topology as well as to researchers in nearby fields. Senior researchers in low-dimensional topology will also be welcome.   

Registration

The school is full and the registration is now closed.