Quantum representations are finite-dimensional projective representations of the mapping class group of a compact oriented surface that arise from the study of Chern--Simons theory; a 3-dimensional quantum field theory. The input to Chern--Simons theory is a compact, connected and simply connected Lie group $G$ (and in my talks, the relevant groups are $G = SU(N)$) and a natural number $k$ called the level. In these talks, I will discuss the representations from two very different and disjoint viewpoints.
<b>Part I: Quantum representations and their asymptotics</b>
The characters of the representations are directly related to the so-called quantum SU(N)-invariants of 3-manifolds that physically correspond to the Chern--Simons partition function of the 3-manifold under scrutiny.
In this talk I will give a definition of the quantum representation using the geometric quantization of the moduli space of flat $SU(N)$-manifolds, where Hitchin's projectively flat connection over Teichmüller space plays a key role. I will give examples of the large level asymptotic behaviour of the characters of the representations and discuss a general conjecture, known as the Asymptotic Expansion Conjecture, for the asymptotics.
Whereas I will likely be somewhat hand-wavy about the details of the construction, I hope to introduce the main objects going into it -- some prior knowledge of the geometry of moduli spaces of flat connections will be an advantage but not necessarily necessary.
<b>Part II: Quantum representations and their algebraic properties</b>
In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.
Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.