Rubber maps and double ramification cycles

The double ramification cycle on the moduli space of marked curves of compact type parametrizes marked, stable, compact type curves admitting a finite degree covering of the projective line with prescribed ramification over zero and infinity. In this talk I will discuss a comparison theorem over the compact type locus between double ramification cycles and classes pushed forward from spaces of rubber stable maps to the moduli space of curves. Recent generalizations of this whole situation will also be discussed. This is joint work with Jonathan Wise and Gabriele Mondello.