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

Solutions to the wave equation preserve energy, and so, in some sense, they maintain their regularity over time. Nevertheless, there is a paradoxical local smoothing phenomenon: When measured in certain aspects, waves can become significantly smoother in a given region of space and time, particularly if one is willing to work “on average” by ignoring some outlier times. Quantifying this phenomenon has been important in applications related to partial differential equations and mathematical physics and has surprising connections to incidence geometry, combinatorics and even number theory. For instance, there is an unexpected connection to the Kakeya needle problem concerning the smallest area needed to rotate a unit line segment (or needle) 180 degrees on a plane.

In this Presidential Lecture, Terence Tao will survey recent and not-so-recent developments in this subject.