Algebraic methods in discrete analogs of the Kakeya problem

We prove the joints conjecture, showing that for any N lines in R³, there are at most points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N² lines in R³ so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N³). Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.