Representations of Kauffman bracket skein algebras of a surface

The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and a representation of the skein algebra features in Witten's topological quantum field theory interpretation of the Jones invariant. Later, the skein algebra and its representations was discovered to bear deep relationships to hyperbolic geometry, via the $SL_2 \mathbb C$-character variety of the surface. This talk will focus on representations of the skein algebra, and particularly how to construct them and how to tell them apart. The latter will involve Chebyshev polynomials and numerous "miraculous cancellations". This is joint work with Francis Bonahon.