Compactified Jacobians and the Double Ramification Cycle

The double ramification cycle -- roughly speaking, the cycle of curves admitting a rational function with prescribed ramification profile -- is an algebraic cycle in the moduli space of curves, intimately connected to Gromov-Witten theory and classical Abel-Jacobi theory. The DR cycle has been extensively studied in recent years; one of the outcomes of this study is a remarkable formula in terms of simple classes in the tautological ring of \bar{M}_{g,n}. However, for certain more delicate questions involving the DR, such as computing its higher dimensional analogues or its behavior under intersection, one must study certain refinements of the DR, for which the existing methods do not give analogous formulas. In this talk I will discuss joint work with Holmes, Pandharipande, Pixton and Schmitt on how one can obtain such formulas by studying the DR via compactified Jacobians.