Appears in collection : Summer School 2024 - Low dimensional Topology

The Witten-Reshetikhin-Turaev quantum representations are (families of) projective representations of mapping class groups of surfaces, with striking properties: the Dehn twists have finite image, but the representations have a unitary structure and infinite images. We will give a construction of the SO(3)-variant of WRT representations as part of the TQFTs underlying the SO(3)-Witten-Reshetikihin-Turaev invariants of 3-manifolds. We will discuss two deep theorems about WRT-representations: asymptotic faithfulness and the density of the image. Finally, we will present some results about the AMU conjecture, which explains how pseudo-Anosov mapping classes are detected by quantum representations. We will also give restrictions on the kernel of quantum representations at fixed level.