# Invariants in low dimensional geometry ad topology

Invariants in low dimensional geometry and topology is a thematic period planned at Institut de Mathématiques de Toulouse from march till july 2017 and financed by CIMI.
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. 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. |

Here is a list of the planned activities:

- A mini-course on Quantum information and quantum computation for mathematicians by Greg Kuperberg, from March the 6th till March the 10th 2017.
- A mini course on Character Varieties and applications to Quantum Topology by Adam Sikora, on March the 20th and the 21st 2017.
- The la Llagonne workgroup on ``L
^{2}-invariants of knots and three manifolds", from March the 27th till March the 31st 2017. - A thematic school on "Quantum topology and geometry", from the 9th of May until the 13th of May 2017
- An international conference on "Quantum topology and geometry in Toulouse", from the 15th of May until the 19th of May 2017
- A thematic school on "Quasi-isometric rigidity" from the 26th of June till the 14th of July 2017
- A weekly Workgroup on Quantization starting on February the 10th

Organisers: Javier Aramayona, Francesco Costantino, Thomas Fiedler, Peter Haïssinsky, Cyril Lecuire