ModHoQTop

The construction of linear representations of mapping class groups of surfaces is notably motivated by the question of the linearity of these groups. In this project, we study two major families of linear representations: homological and quantum representations, which provide strong candidates for being faithful and thus advance the question of linearity. Quantum representations, in particular, are part of TQFTs (Topological Quantum Field Theories) and are thus connected to families of topological invariants for all three-dimensional embeddable manifolds. They are based on highly algebraic constructions, and a key issue is to understand what they can topologically detect. This project, therefore, consists of three major axes: - Studying the two families of representations, particularly trying to find faithful ones. There are two strategies: a homological one, which aims to extend Bigelow's successful proof in the case of discs, and a quantum one that would generalize the Freedman–Walker–Wang proof of asymptotic faithfulness for semi-simple TQFTs. - Fully reconstructing TQFTs homologically by continuing work already begun at the level of braids, knots, and representations of mapping class groups. Homologically interpreting quantum invariants would allow for a better understanding of what they topologically detect, thereby making better use of their richness. - Reconstructing quantum groups and skein algebras homologically. These are the algebraic structures at the core of TQFT construction. The richness of their representation theory is also a source of mysteries that a geometric-topological reconstruction could help clarify.