Logarithmic bounds for isoperimetry and slices of convex sets

We prove that the Bourgain slicing conjecture and the Kannan-Lov’asz-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 a refined log-concave Lichnerowicz inequality.