  We give the first non-trivial upper bounds on the average sensitivity and
noise sensitivity of polynomial threshold functions. More specifically, for a
Boolean function f on n variables equal to the sign of a real, multivariate
polynomial of total degree d we prove
  1) The average sensitivity of f is at most O(n^{1-1/(4d+6)}) (we also give a
combinatorial proof of the bound O(n^{1-1/2^d}).
  2) The noise sensitivity of f with noise rate \delta is at most
O(\delta^{1/(4d+6)}).
  Previously, only bounds for the linear case were known. Along the way we show
new structural theorems about random restrictions of polynomial threshold
functions obtained via hypercontractivity. These structural results may be of
independent interest as they provide a generic template for transforming
problems related to polynomial threshold functions defined on the Boolean
hypercube to polynomial threshold functions defined in Gaussian space.
