Abstract: This work constitutes a very preliminary step in the
study of geometric heat flows on stratified spaces. It concerns the
one-dimensional case of singular networks. Work on this goes back
several decades (motivated by metallurgy!), and I will review some of
the main points of progress, and then discuss a new formulation of the
problem which clarifies short-time existence around "irregular
vertices", and is expected to be particularly useful for the higher
dimensional version. This is joint work with Jorge Lira, Alessandra
Pluda and Mariel Saez.