Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials. 

The BGG category O plays an important role in the study of representations of semisimple Lie algebras. Its connection to the Hecke category is a starting point of Geometric Representation Theory. In this lecture, I will introduce a version of category O for quantum groups at roots of unity. I will explain a derived equivalence from (the principal block of) quantum category O to the affine Hecke category. Under the equivalence, the highest weight structure of quantum category O provides a categorification of the “periodic Hecke module”.

In the third lecture, I will introduce the main result on an equivalence between quantum category O and affine Hecke category. Then I will explain the corresponding t-structure on the affine Hecke category, and its relation to the periodic Hecke module.