Szego Seminar: The surprising bridge between geometry and maximal functions 

Abstract: A Kakeya set is a subset of Rn that contains a unit line segment in every direction. It turns out that understanding these sets has many applications in Fourier analysis and other important related fields. A natural question is how small a Kakeya set can be. Besicovitch showed that there exists Kakeya sets with Lebesgue measure zero in every dimension... quite the surprise! This led to the Kakeya conjecture, which states that a Kakeya set in Rn has (Hausdorff) dimension n. Essentially, this is saying that although a Kakeya set may be null in the eyes of Lebesgue measure, there is still enough mass that it must be an n-dimensional object. There are many ways to approach this problem, but one particularly interesting one is linking this statement to the Lp boundedness of a geometric maximal function. We will explore this connection and see that this is a general principle for Kakeya-like sets.   

Host: Pooja Joshi