The APDE seminar on Monday, 2/27, will be given by Joshua Zahl (University of British Columbia) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Mengxuan Yang (firstname.lastname@example.org).
Title: Sticky Kakeya sets, and the sticky Kakeya conjecture
Abstract: A Kakeya set is a compact subset of 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.