Triangular decomposition of skein algebras

The Kauffman bracket skein modules are the natural habitat for the Jones polynomial of links in oriented 3-manifolds. If the 3-manifold is a surface times an interval, the skein module carries an interesting but poorly understood associative algebra structure, with multiplication given by stacking along the interval direction. The resulting Kauffman bracket skein algebras give quantizations of $SL(2,C)$ character varieties and have relations to quantum cluster algebras and quantum Teichmüller space. The goal of the talk is to get a better understanding of these objects and relationships from the viewpoint of a relatively recent paper of Thang Lê that introduces stated skein algebras for surfaces, which can be decomposed along an ideal triangulation.