The Atiyah-Singer index theorem is a paramount result which depicts an interplay between Analysis, Topology and Geometry. Although its classical formulation takes place in the context of compact manifolds without boundary, subsequent developments have aimed to extend it to broader classes of spaces. One of several lenses through which to analyse the problem, the so-called heat kernel method, seems to offer a way of looking for an analogue in singular spaces, for which there are many natural and relevant examples (algebraic varieties, moduli spaces, etc.).

In this talk, I would like to briefly discuss how this method can be adapted for a certain class of singular manifolds whose boundary is the total space of a fibration. The key ingredient will be a blow-up analysis à la Melrose of the asymptotics of the heat kernel.