We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that ifPNP[1℄ = PNP[2℄, then the polynomial-time hierarchy collapses to Sp2 p2 \ p2. Even showing that the hierarchy collapsed to p2 remained open prior to this paper.

In [1], Airault, McKean and Moser observed that the motion of poles of a rational solution to the K-dV or Boussinesq equation obeys the Calogero-Moser dynamical system [2, 3, 4] with an extra condition on the configuration of poles. In [8], Krichever observed that the motion of poles of a solution to the KP equation which is rational in t1 obeys the Calogero-Moser dynamical system. In this note...

McKean and Trubowitz [2] showed that the theory of the KdV equation ∂ ∂t g(x, t) = ∂ 3 ∂x 3 g(x, t) − 6g(x, t) ∂g ∂x (x, t). is intimately related to the geometry of a related hyperelliptic curve of infinite genus, the Bloch spectrum B g t of the operator L g t : ψ → d 2 dx 2 ψ(x) + g(x, t)ψ(x), where g t = g(x, t). As was known classically, B g t is independent of t, when g(x, t) evolves accor...