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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

March 2, Noon Pacific, Colloquium by Larry Guth (MIT)

An enemy scenario in restriction theory

Abstract: The goal of this talk is to explore some of the difficulties involved in an open problem in restriction theory. The problem is the Mizohata-Takeuchi conjecture for the parabola, which is a special case of a conjecture Stein raised. This is a conjecture about weighted estimates for the extension operator for the parabola. While we focus on this problem for concreteness, I think that there are similar difficulties involved in a number of open problems in restriction theory.

A lot of work in restriction theory leans heavily on wave packet decompositions, and on a short list of basic properties of wave packet decompositions. For instance, the proof of decoupling for the parabola (or paraboloid or moment curve) only uses this list of basic properties. We will describe an enemy scenario for the M-T conjecture. This scenario would respect the basic properties of wave packet decompositions used in decoupling, but would violate the M-T conjecture. I don't know whether this scenario actually happens. In order to rule out this scenario, we would have to find a new tool.