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

  • Speaker
  • Portrait photo of Terence TaoTerence Tao, Ph.D.Professor, University of California, Los Angeles
Date & Time

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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.

About the Speaker

Portrait photo of Terence Tao

Tao was born in Adelaide, Australia, in 1975. He has been a professor of mathematics at the University of California, Los Angeles, since 1999, having completed his Ph.D. under Elias Stein at Princeton University in 1996. Tao’s research areas include harmonic analysis, partial differential equations, combinatorics and number theory. He has received many awards, including the Salem Prize in 2000, the Bochner Prize in 2002, the Fields Medal in 2006, a MacArthur Fellowship in 2007, the Waterman Award in 2008, the Nemmers Prize in 2010, the Crafoord Prize in 2012 and the Breakthrough Prize in mathematics in 2015. Tao also holds the James and Carol Collins chair in mathematics at UCLA and is a fellow of the Royal Society, a corresponding member of the Australian Academy of Sciences, a foreign member of the National Academy of Sciences and a member of the American Academy of Arts and Sciences.

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