The tautological ring of Mgn-bar has been a crucial object in the study of the intersection theory of the moduli space of curves. Recently, there has been more focus on the logarithmic enumerative geometry of Mgn-bar, with interesting classes coming from e.g. log double ramification cycles. We present a definition of the log tautological ring of Mgn-bar, together with a log decorated strata algebra, and prove several structure results. The main new tools are the notions of cone stacks with boundary and homological piecewise polynomials, that capture the tropicalisation of strata of log smooth stacks and the combinatorial part of their intersection theory.

This is joint work with Rahul Pandharipande, Dhruv Ranganathan and Johannes Schmitt.