Jacobians, Stable Maps, and Cycles on the Moduli Space of Curves

The cohomology ring of the moduli space of curves contains two families of classes with different geometric origins: one comes from the universal Jacobian and Brill-Noether theory, and the other from the moduli space of stable maps and Gromov-Witten theory. After reviewing their construction, I will explain how new ideas from logarithmic geometry connect these classes, and discuss how the connection allows us to explicitly calculate them in the tautological ring, in terms of its standard generators. Time permitting, I will indicate the implications these calculations have for the study of relations in the tautological ring. The talk will draw from several joint works and some work in progress, carried out with Holmes-Pandharipande-Pixton-Schmitt, Abreu-Pagani, Ranganthan, and Wise.