Generalized boundary strata classes

The Deligne-Mumford moduli space of stable curves of genus g admits a stratification by topological type.
These boundary strata define classes in the Chow ring of the moduli space of stable curves. I will describe
a natural generalization of these boundary strata classes to unstable topological types. I will then explain
how to use the generalized boundary strata classes to define a small subring of the Chow ring containing
every double ramification cycle.