Quantum Representations of Mapping Class Groups and Their Applications

Given sufficiently nice algebraic data (for our purposes a modular tensor category) it is possible to construct a (2+1) TQFT, which is a compatible assignment of vector spaces to surfaces.  This compatibility ensures that each vector space admits an action of the mapping class group of the corresponding surface.  These families of representations are called quantum representations of mapping class groups.  In the case that we will be discussing the underlying state space given to surfaces will be constructed out of colored ribbon graphs.  We will look at these quantum representations and describe some applications both within low dimensional topology and also to topological quantum computation.  This is joint work with Zhenghan Wang.