Quantum Rings and Recursion Relations in 2D Quantum Gravity
  I study tachyon condensate perturbations to the action of the two dimensional string theory corresponding to the c=1 matrix model. These are shown to deform the action of the ground ring on the tachyon modules, confirming a conjecture of Witten. The ground ring structure is used to derive recursion relations which relate (N+1) and N tachyon bulk scattering amplitudes. These recursion relations allow one to compute all bulk amplitudes. 
Shamit Kachru