  We prove $d$-linear analogues of the classical restriction and Kakeya
conjectures in $\R^d$. 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 $n<d$.
