  In this paper we establish new geometric and analytic bounds for Ricci flows,
which will form the basis of a compactness, partial regularity and structure
theory for Ricci flows in [Bam20a, Bam20b].
  The bounds are optimal up to a constant that only depends on the dimension
and possibly a lower scalar curvature bound. In the special case in which the
flow consists of Einstein metrics, these bounds agree with the optimal bounds
for spaces with Ricci curvature bounded from below. Moreover, our bounds are
local in the sense that if a bound depends on the collapsedness of the
underlying flow, then we are able to quantify this dependence using the pointed
Nash entropy based only at the point in question.
  Among other things, we will show the following bounds: Upper and lower volume
bounds for distance balls, dependence of the pointed Nash entropy on its
basepoint in space and time, pointwise upper Gaussian bound on the heat kernel
and a bound on its derivative and an $L^1$-Poincar\'e inequality. The proofs of
these bounds will, in part, rely on a monotonicity formula for a notion, called
variance of conjugate heat kernels.
  We will also derive estimates concerning the dependence of the pointed Nash
entropy on its basepoint, which are asymptotically optimal. These will allow us
to show that points in spacetime that are nearby in a certain sense have
comparable pointed Nash entropy. Hence the pointed Nash entropy is a good
quantity to measure local collapsedness of a Ricci flow
  Our results imply a local $\varepsilon$-regularity theorem, improving a
result of Hein and Naber. Some of our results also hold for super Ricci flows.
