Abstract:
In 1919, Besicovitch constructed a compact set in the plane with Lebesgue measure 0 that contains a unit line segment pointing in every direction. Such objects are now called measure 0 Besicovitch sets (aka Kakeya sets). By replacing a measure zero Besicovitch set by its \delta-thickening, one obtains a collection of 1 x \delta rectangles pointing in different directions, the sum of whose areas is 1, but whose union has very small volume. The existence of such collections of rectangles is called the Besicovitch compression phenomenon.
The Kakeya set conjecture is a quantitative statement controlling the strength of the Besicovitch compression phenomenon. In this talk, I will discuss connections between the Besicovitch compression phenomenon, the Kakeya set conjecture, and questions in harmonic analysis and PDE.

Biography

Prof. Joshua Zahl is a chair professor at the Chern Institute of Mathematics, Nankai University. He obtained a Ph.D. from the University of California, Los Angeles in 2013 under the supervision of Prof. Terence Tao. Before joining Nankai University in 2025, he was an associate professor at the University of British Columbia, where he solved the famous Kakeya conjecture in three dimensions joint with Prof. Hong Wang. He has published numerous articles in prestigious journals such as Inventiones Mathematicae, Journal of the American Mathematical Society, Duke Mathematical Journal, Geometric and Functional Analysis and the American Journal of Mathematics.