  We present a monotonic expression for the Ricci flow, valid in all dimensions
and without curvature assumptions. It is interpreted as an entropy for a
certain canonical ensemble. Several geometric applications are given. In
particular, (1) Ricci flow, considered on the space of riemannian metrics
modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is,
other than fixed points); (2) In a region, where singularity is forming in
finite time, the injectivity radius is controlled by the curvature; (3) Ricci
flow can not quickly turn an almost euclidean region into a very curved one, no
matter what happens far away. We also verify several assertions related to
Richard Hamilton's program for the proof of Thurston geometrization conjecture
for closed three-manifolds, and give a sketch of an eclectic proof of this
conjecture, making use of earlier results on collapsing with local lower
curvature bound.
