We look at bond percolation on “graph blowups” of a graph G. This is a generalization of the Erdos-Renyi random graph, and it features an analogous phase transition with respect to the appearance of a giant component. We show that the vector multiplicities of the giant component converge (after suitably centering and rescaling) to a Gaussian field on G whose covariance can be computed explicitly as the square of a massive Green’s function on G. The proof strategy is combinatorial and relies on the combinatorics of spanning trees on blowup graphs, whose generating function has a beautiful analytic structure.

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.