Pointwise convergence for the Schr\"odinger equation [After Xiumin Du and Ruixiang Zhang]

This expository essay accompanied the author's presentation at the S\'eminaire Bourbaki on 01 April 2023. It describes the breakthrough work of Du--Zhang on the Carleson problem for the Schr\"odinger equation, together with background material in multilinear harmonic analysis.Comment: 72 pages, 6 figures, comments welcome

The Fourier restriction and Kakeya problems over rings of integers modulo N

The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting

A note on Fourier restriction and nested Polynomial Wolff axioms

This note records an asymptotic improvement on the known $L^p$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial partitioning method with recent geometric results regarding intersections of tubes with nested families of varieties.Comment: 24 pages, 0 figures. This article builds upon arXiv:1807.1094

Improved Fourier restriction estimates in higher dimensions

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.Comment: 43 pages, 5 figures. A number of typos have been corrected and additional points of clarification added. To appear in Cambridge Journal of Mathematic

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.Comment: 27 pages, 3 figures. Update incorporates referee suggestions. Minor correction to the proof of Lemma 2.6. To appear in APD