in this research community
Fourier restriction conjecture and related problems
American Institute of Mathematics, San Jose, California
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 firstname.lastname@example.org
Upcoming activitiesTuesday September 26 at 1:00-3:00pm Pacific, Colloquium by Joshua Zahl (UBC)
Sticky Kakeya sets, and the sticky Kakeya conjecture
Abstract: A Kakeya set is a compact subset of $\mathbb R^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have dimension $n$. This conjecture is closely related to several open problems in harmonic analysis, and it sits at the base of a hierarchy of increasingly difficult questions about the behavior of the Fourier transform in Euclidean space. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate self-similarity at many scales, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. In this talk, I will discuss a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have dimension $n$. I will discuss the proof of this conjecture in dimension 3.
This is joint work with Hong Wang.