Short description
The emergence of topological quantum field theory in mathematics can be traced to the work of E. Witten at the end of the 80's. Prior to this M.Atiyah, N.Hitchin and S.Donaldson had shown that Yang-Mills Theory (the standard model of particles) was of profound mathematical interest providing a powerful tool to analyse the geometry of four dimensional manifolds. The success of their approach was breathtaking. The Witten's results added force to the notion that one can obtain results in low dimensional topology by applying techniques from quantum field theory. Broadly speaking, the underlying principal evoked in their work is that topology in dimensions three and four should be considered as a branch of physics, more particularly of quantum field theory. Following this realisation, a cascade of results were proven and new directions of research were opened, giving rise to what one might call the ``quantum world'': quantum groups, quantum invariants etc. Our philosophy in this project, is to apply techniques and lessons of the quantum approach to low dimensional topology more widely in the study of actions and representations of the mapping class groups of surfaces. A fundamental difficulty in the study of the the mapping class group is that we know very few linear representations. The mapping class group does admit natural actions on spaces of structures related to the surface; for example the space of laminations of the surface, the curve and arc complexes, and the character variety of the fundamental group.Our aim is to study these actions together with the actions on the associated quantifications of the underlying spaces, exploiting the interplay between geometry, topology and dynamics, leading to new perspectives in the theory of the mapping class group.