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Fourier restriction conjecture and related problems

An online research community sponsored by the

 

American Institute of Mathematics, San Jose, California

organized by

Dominique Maldague, Yumeng Ou, Po-Lam Yung, and Ruixiang Zhang

This research community, sponsored by AIM and the NSF, is for mathematicians studying Fourier restriction theory and related problems. Originating from a deep observation of Stein in 1967, the Fourier restriction conjecture predicts that if $S$ is a curved smooth submanifold of Euclidean space, then taking the Fourier transform and restricting to S extends to a bounded linear operator (from $L^p(\mathbb R^n)$ to $L^1(S)$, for certain $p$). This is somewhat unexpected since the Fourier transform of an $L^p$ function is a priori only almost everywhere defined and $S$ is a zero measure set in $\mathbb R^n$. This conjecture is deep and far-reaching and has surprising connections to PDE, number theory, incidence geometry, combinatorics and geometric measure theory.

Our research community hosts small working groups, occasional larger problem sessions and seminars, and periodic social events. Our main aim is to introduce earlier career mathematicians to the field and to the community, as well as to facilitate collaborations within the field.

If you are interested in joining our research community, please fill out this application form. Applications are open to all, and we especially encourage women, underrepresented minorities, and researchers from primarily undergraduate institutions to apply.

For more information, email research@aimath.org

Upcoming activities

Friday September 16 at 2:00 Pacific, Colloquium by Hong Wang (UCLA)

Restricted projections to planes in $\mathbb{R}^3$

Abstract: Let $\gamma: [0,1]\rightarrow \mathbb{S}^2$ be a non-degenerate curve and let $P_{\theta}: \mathbb{R}^3 \rightarrow \gamma(\theta)^{\perp}$ be the orthogonal projection. We show that if $A\subset \mathbb{R}^3$ is a Borel set, then for almost all $\theta\in [0,1]$, $\dim_H P_{\theta} A = \min \{ 2, \dim_H A\}$ . This confirms the second part of a conjecture by Faessler and Orponen. The proof uses Fourier analysis for incidences for tubes.

This is joint work with Gan, Guo, Guth, Harris and Maldague.