How Big Can It Be? Quantifying Size in Fourier Analysis

Abstract. In this talk I will discuss a few problems of quantifying the notion of size in the mathematical area of Fourier analysis. The fundamental issue is that in essentially any sufficiently complex system, there are multiple “natural” ways to understand or quantify the notion of size. This leads to a never-ending series of questions in comparing different notions, like: does largeness in one sense always lead to largeness in the other sense? The main part of the talk will be about the Kakeya Needle Problem, which examines whether sets which are large enough to move a needle-shaped object around inside must also be large in the usual sense of area. This problem has an interesting and satisfying solution, but is also intimately connected to a host of open questions, large and small, in Fourier analysis. As time permits, we will explore connections to geometric nonconcentration inequalities, which are a general framework for figuring out how to define largeness of sets so that it corresponds with whatever geometric properties that you find interesting.

Brief Biography. Dr. Philip T. Gressman is a Professor of Mathematics at University of Pennsylvania. He received his BA from Washington University in St. Louis in 2001 and his PhD in harmonic analysis in 2005 from Princeton University, advised by Elias M. Stein. From 2005 to 2008, he was a Gibbs Assistant Professor at Yale University, and he has been a member of the Penn math faculty since 2008. Gressman’s work mixes elements of geometry, partial differential equations, and Fourier analysis and has been supported by the National Science Foundation since 2006. In 2011, he received an Alfred P. Sloan Research Fellowship for his joint work with Robert M. Strain, in which they were the first to prove existence, uniqueness, and rapid convergence to equilibrium of perturbative solutions of the Boltzmann equation, a result which had eluded proof for roughly 140 years. He is also dedicated to undergraduate teaching. He is currently the Undergraduate Chair of the Mathematics Department and was the 2020 recipient of Penn’s College of Arts and Sciences Dennis M. DeTurck Award for Innovation in Teaching.