With T. Nevins. **Toda lattice hierarchy and noncommutative geometry**.

We apply the algebraic geometry of additive, multiplicative and
elliptic difference operators using generalizations of the Fourier-Mukai transform.
Namely we construct moduli spaces of difference modules
on (cuspidal, nodal or smooth) cubic curves and identify them with moduli
spaces of spectral sheaves on C^* bundles over the same curves.
We then apply this description to a problem of integrable systems explored
by Krichever and Zabrodin: the relation
between rational, trigonometric and elliptic solutions of the Toda lattice hierarchy
(and its nonabelian generalizations) and the Ruijsenaars-Schneiders (RS) particle
systems (and their spin variants). We show that the Toda lattice has a natural
interpretation as flows on difference modules and that the Fourier-Mukai transform
puts the Toda hierarchy in action-angle form, identifying the
difference modules with the spectral curves of the RS systems and the poles of
the Toda solutions with the positions of the RS particles.