On the Multilinear Restriction and Kakeya conjectures

We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in ${\mathbb{R}}^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called “joints” problem, as well as presenting some $n$-linear analogues for $n<d$.