The physical properties of materials are often described by singular integrals over the Brillouin zone, a structure with the topology of a torus. In this talk, we will introduce the quantum formalism and the types of integrals of interest, with a particular focus on the simple case of the Density of States (DOS). 

First, we will establish the framework for these integrals and outline the computational challenges they present. We will then review standard computational methods for evaluating these integrals and discuss recent advancements. Finally, we will present two recent approaches: the Brillouin Contour Deformation (BCD) technique, and the implicit integration method. Through their application to DOS computations, we demonstrate the efficiency of both methods while highlighting the specific challenges associated with this class of integrals.