Abstract: Joints and multijoints provide discrete analogues of the Kakeya maximal function and multilinear Kakeya respectively. While Guth's sharp endpoint multilinear Kakeya theorem in Euclidean space is established "on the side of the maximal function", Zhang's joint and multijoint theorems are established "on the side of the covering lemma". We explore the dualities between these alternative approaches, both in the context of joints/multijoints and also more abstractly. This is joint work with Michael Tang.

Abstract: In this talk, which will be based on joint research with S. Buschenhenke and A. Vargas, I intend to discuss some of the new challenges that arose in our studies of Fourier restriction estimates for hyperbolic surfaces, compared to the case of elliptic surfaces.

Given the complexity of the bilinear, and even more so of the polynomial partitioning approach, I shall mainly focus on those parts of these methods which required new ideas, so that a familiarity with these methods will not be expected from the audience.

Abstract: Let F be a finite subset of Z^d. We say that F is a translational tile of Z^d if it is possible to cover Z^d by translates of F without any overlap. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, asserts that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some recent results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.

Abstract: This is joint work with Olli Saari. We will review some highlights of the theory of Fourier multipliers in one dimension, such as Coifman-Rubio-de-Francia Semmes theory, and variational Carleson estimates. We will then discuss two dimensional multiplier theorems, in particular multipliers which are characteristic functions of convex sets. We present some new results and some open problems.

Abstract: Decoupling and discrete restriction inequalities have been very fruitful in recent years to solve problems in additive combinatorics and analytic number theory. In this talk I will present some work in decoupling for Cantor sets, including Cantor sets on a parabola, decoupling for product sets, and give applications of these results to additive combinatorics. Time permitting, I will present some open problems.

Contains joint work with Alan Chang (Princeton), Rachel Greenfeld (UCLA), Asgar Jamneshan (Koç University), Zane Li (IU Bloomington), José Ramón Madrid Padilla (UCLA).