From Rotating Needles to Stability of Waves: Local Smoothing in the 21st Century

Explore a fascinating 46-minute lecture that delves into the paradoxical local smoothing phenomenon of wave equations and their preservation of energy. Learn how waves can become significantly smoother in specific regions of space and time when measured in certain aspects, particularly when working "on average" by disregarding outlier times. Discover the important applications of this phenomenon in partial differential equations and mathematical physics, along with its surprising connections to incidence geometry, combinatorics, and number theory. Gain insights into the unexpected relationship with the Kakeya needle problem, which examines the minimal area required to rotate a unit line segment 180 degrees on a plane. Through this Presidential Lecture at the Simons Foundation, gain a comprehensive understanding of both recent and historical developments in this intriguing mathematical subject.