A new construction of $c=1$ Virasoro conformal blocks

The Virasoro conformal blocks are very interesting since they have many connections to other areas of math and physics. For example, when $c=1$, they are related to tau functions of integrable systems of Painlev\'{e} equations. They are also closely related to non-perturbative completions in the topological string theories. I will first explain what Virasoro conformal blocks are. Then I will describe a new way to construct Virasoro blocks at $c=1$ on $C$ by using the "abelian" Heisenberg conformal blocks on a branched double cover of C. The main new idea in our work is to use a spectral network and I will show the advantages of this construction. This nonabelianization construction enables us to compute the harder-to-get Virasoro blocks using the simpler abelian objects. It is closely related to the idea of nonabelianization of the flat connections in the work of Gaiotto-Moore-Neitzke and Neitzke-Hollands. This is based on a joint work with Andrew Neitzke.