Workshop: Number-theoretic methods to produce counterexamples for questions motivated by PDE’s, Lillian Pierce

Speaker
Lillian Pierce, Duke University

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 seemed to gain its power from simple properties of Gauss sums. In this talk we will explain how Bourgain’s counterexample works, and describe a new, flexible way to use the Weil bound to create counterexamples relevant to large classes of dispersive PDE’s. We will end with some open questions.