The problem of computing scl in free product is to find the most
efficient surface mapping into a wedge of spaces subject to a restriction
on the boundary map. The difficulty is that the space of surface mappings
is huge. I will present an idea of Danny Calegari to decompose the surface
and use a FINITE dimensional polyhedral cone to encode a small amount of
the information to compute scl. Similar ideas appear in geometry/topology
quite often. I will explain how to get rationality and spectral gap (if
time permits) from this.