Logarithmic bounds for isoperimetry and slices of convex sets

We prove that the Bourgain slicing conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture in ${\mathbb{R}}^n$ hold true up to a factor of $\sqrt{\log n}$. A new ingredient used in the proof is an improved log-concave Lichnerowicz inequality.