The polynomial method in Kakeya-type problems

Ruixiang Zhang, Princeton University
Fine Hall 314

In the past several years, the so-called "polynomial method" have been used to prove several problems related to analysis and combinatorics. By modeling the given objects (e.g. points or small cubes) with a polynomial, this method enables people to solve a bunch of challenging problems having the flavor of incidence geometry. The most remarkable part of this new perspective is that it often leads to short proofs of theorems that would otherwise seem quite hard. Among such problems, the Kakeya-type problems are of special interest because of the importance of the unsolved Kakeya conjectures. We will talk about three examples of applications of the polynomial method to Kakeya-type problems: Dvir's proof of the finite field Kakeya theorem, proofs of the endpoint Bennett-Carbery-Tao multilinear Kakeya theorem by Guth and Carbery-Valdimarsson, and the joints theorem proved by Guth-Katz, Quilodran and Kaplan-Sharir-Shustin. If time permitted, we will also talk about a recent work on a discrete model of the Kakeya and Furstenberg problems in R^n.