# Applications of Multilinear Restriction and Restriction Estimates

# Applications of Multilinear Restriction and Restriction Estimates

The restriction problem is an important open problem in harmonic analysis. The 2-dimensional case was proven in the 1970's. The case of three (or more) dimensions remains open, with interesting partial results. In 2005, Bennett, Carbery, and Tao proved a ``multilinear" restriction estimate. We can think of this work as a near-optimal estimate for a significant chunk of the terms in the original restriction problem. It had major philosophical impact, giving a new perspective on what part of the problem is most difficult. Recently, the multilinear estimate has also had some practical impact. A paper by Bourgain and me uses the multilinear inequality to prove estimates for the original restriction problem, matching and sometimes improving the best estimates previously known. I will briefly review the restriction problem, explain the multilinear restriction theorem, and explain how to apply it to the 3-dimensional restriction problem.