Abstract: Inside the Jacobian of the universal curve of the moduli
space of smooth curves lie the Brill-Noether loci, parametrizing pairs
of a smooth curve with a line bundle that has more than expected
sections. Intersecting these cycles with any section of the universal
Jacobian produces a family of cycles on the moduli space, which lie in
the so-called tautological subring of its Chow ring, and which
determine several of its structural properties. Over the
Deligne-Mumford compactification of the moduli space by stable curves,
the meaning of these cycles is far more subtle, owing essentially to
the fact that the compactification of the Jacobian over the moduli
space of stable curves is itself a subtle problem.  Using the theory
of compactified Jacobians, Pagani, Ricolfi and van Zelm have proposed
an extension of these cycles to the moduli space of stable curves, and
conjecture that they also lie in the tautological ring. In this talk,
I will describe a natural theory of tautological rings of compactified
Jacobians, which streamlines such questions and allows us to prove
that the PRvZ conjecture holds.