Invariants in low dimensional geometry ad topology

Low-dimensional topology and geometry are inextricably linked and faced an impressive growth in the last thirty years. The main focus of this semester will be quantum invariants of links and three-manifolds and quasi-isometric invariants of groups.

Quantum invariants of links and 3-manifolds are natural byproducts of TQFTs (Topological Quantum Field Theories). One of the main examples of these invariants are the Reshetikhin-Turaev invariants whose definition, based on the representation theory of quantum groups, mixes remarkably topology and algebra. These invariants provide in particular new and interesting invariants of knots of which the famous Jones polynomials are the main examples.
The recent developments are increasingly putting forward the roles played by hyperbolic geometry and by arithmetics in the study of these invariants. For instance Kashaev’s volume conjecture postulates a deep and unexpected link between the Jones polynomial and the Gromov norm of a knot complement and the so-called AJ- conjecture predicts a link between some recursion relations (expressed in terms of q-difference equations) satisfied by the Jones polynomials of knots and the variety of representations in SL2(C) of the fundamental group of the complements of the knots. This area, nowadays called "Quantum Topology", will be explored in the context of the present semester via a thematic school and a conference (see below). Furthermore this year's la Llagonne workgroup will be focused on L² invariants of knots and three manifolds, a topic linking group theory to quantum topology and hyperbolic geometry via the relations of the L² -torsion with hyperbolic volume. 

On the geometric group theory side, an important number of questions in low-dimensional topology may be interpreted in terms of the study of properties of groups via geometric methods. A central idea in this direction is an observation of Svarc and Milnor, who noted that a discrete group acting cocompactly on a proper space is, in coarse-geometric terms, equivalent to the given space; the works of Thurston and Gromov in the 1970s and 1980s respectively gave a major boost to this topic. One of the central problems in geometric group theory, originally due to Gromov is the classification of finitely generated groups up to quasi-isometry. While completely open in its full generality, this question has seen major breakthroughs for important classes of groups and spaces. A survey of the most important results, as well as the main techniques and ideas involved in the proofs, will be the central theme of the 3-week summer school.

In accordance with the Statement of Inclusiveness , this event will be open to everybody, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity.