  The one dimensional Kardar-Parisi-Zhang universality class is believed to
describe many types of evolving interfaces which have the same characteristic
scaling exponents. These exponents lead to a natural renormalization/rescaling
on the space of such evolving interfaces. We introduce and describe the
renormalization fixed point of the Kardar-Parisi-Zhang universality class in
terms of a random nonlinear semigroup with stationary independent increments,
and via a variational formula. Furthermore, we compute a plausible formula the
exact transition probabilities using replica Bethe ansatz. The semigroup is
constructed from the Airy sheet, a four parameter space-time field which is the
Airy2 process in each of its two spatial coordinates. Minimizing paths through
this field describe the renormalization group fixed point of directed polymers
in a random potential. At present, the results we provide do not have
mathematically rigorous proofs, and they should at most be considered
proposals.
