Abstract: It has been known since the work of Airault, McKean, and Moser in 1977 that the time evolution of poles for meromorphic solutions to the Kadomtsev-Petviashvili (KP) equation are related to dynamics of Calogero-Moser (CM) particles. David Ben-Zvi and Tom Nevins proved a general result on a correspondence between the full KP and CM hierarchies in arXiv:math/0603720. Their approach is to identify the CM and KP phase spaces with algebro-geometric data on a cubic curve, and relate the two phase spaces via a Fourier-Mukai transform. In this talk I will describe ongoing work, joint with David and Tom, applying the same ideas to prove a correspondence between the Ruijsenaars-Schneider (RS) and 2D Toda lattice hierarchies. I will focus on giving an algebro-geometric description of the spectral curves of the RS system, the proof of which uses identities for theta functions known since the 19th century.