  We adapt Guth's polynomial partitioning argument for the Fourier restriction
problem to the context of the Kakeya problem. By writing out the induction
argument as a recursive algorithm, additional multiscale geometric information
is made available. To take advantage of this, we prove that direction-separated
tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether,
this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$
with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an
infinite sequence of dimensions.
