2023 Excelsior Lectures

Talk 1: Bad behavior

Abstract: What do you do with a person who behaves in the worst possible way at every point in time? Well, I don’t know. But if you ask instead about a mathematical operator that picks out the worst possible behavior of a function at every point, we sometimes know how to control it. In fact, such operators are very useful in harmonic analysis, and have been studied in various guises over the past century. We will survey several classical operators of this kind, and then will explore a few modern questions that involve bad behavior.

Talk 2: Number-theoretic methods to produce counterexamples for questions motivated by PDE’s

Abstract: In 1980 Carleson posed a question in PDE's: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated question was recently resolved by a combination of two results: one by Bourgain, whose counterexample construction proved a necessary condition, and later a complementary result of Du and Zhang, who proved a sufficient condition. 

Bourgain’s counterexample was particularly interesting for two reasons: first, it generated a necessary condition that contradicted what everyone had expected, and second, it was a number-theoretic argument. In this talk we will describe how number theory plays a role, first in the context of Bourgain’s counterexample. Then we will describe a new, far more flexible number-theoretic method for constructing counterexamples, which opens the door to studying convergence questions for other dispersive PDE’s, where many questions remain open. 

About the Excelsior Lecturer

Lillian Pierce, Professor of Mathematics at Duke University,  is a leading researcher at the crossroads of harmonic analysis and number theory.  She earned a B.A. in Mathematics at Princeton, graduating as class valedictorian, and a Ph.D.  in 2009, also  at Princeton, for which her advisor was Elias Stein. She spent 2009-2010 at the Institute for Advanced Study (IAS), and then 2010-2013 at the Mathematical Institute at Oxford, supported by NSF Postdoctoral and Marie Curie International fellowships. These were followed by a von Neumann Fellowship back at IAS in 2017-18 and then a Bonn Research Fellowship at the Hausdorff Center in Bonn (2017-21). Prof. Pierce was the recipient of a Sloan Research Fellowship in 2018, a Presidential Early Career Award in 2019,   Simons  and Guggenheim Research Fellowships in 2023, and she was elected a Fellow of the AMS in 2021.