# A restriction estimate using polynomial partitioning

# A restriction estimate using polynomial partitioning

The restriction conjecture is an open problem in Fourier analysis first raised by Stein in the late 60’s. It is about the L^p estimates obeyed by an oscillatory integral operator. I will explain a recent approach to this problem, which gives a slightly better estimate in three dimensions. The polynomial partitioning approach is a divide-and-conquer argument, using a polynomial surface to cut space into pieces and an inductive argument to understand the contribution of each piece. This approach is based on recent ideas from combinatorics. In 2007, Dvir used polynomials in a surprising way to give a very short proof of the finite field Kakeya problem - a kind of cousin of the restriction problem. Building on these ideas, Katz and I used polynomial partitioning to prove new estimates in incidence geometry. This approach to the restriction problem is a variation on these arguments from combinatorics.