ABSTRACT: Given a Riemann surface C and a central charge c, one can define the notion of Virasoro conformal block. Virasoro conformal blocks capture universal features of conformal field theory on C. I will describe a new scheme for constructing Virasoro conformal blocks at central charge c=1, by relating them to simpler “abelian” objects, namely conformal blocks for the Heisenberg algebra, on a branched double cover of C. The key new ingredient is a spectral network on the surface C. Some particularly important Virasoro conformal blocks at c=1 are also known as “tau functions”, and I will explain what abelianization tells us about them. This is joint work in progress with Qianyu Hao, inspired by work of Coman-Longhi-Pomoni-Teschner, Iwaki, Marino and others.