Emmy Noether Lectures

2018 Lecturer: Jill Pipher

Nonsmooth boundary value problems

The regularity properties of solutions to linear partial differential equations in domains depend on the structure of the equation, the degree of smoothness of the coefficients of the equation, and of the boundary of the domain. Quantifying this dependence is a classical problem, and modern techniques can answer some of these questions with remarkable precision. For both physical and theoretical reasons, it is important to consider partial differential equations with non-smooth coefficients. We’ll discuss how some classical tools in harmonic and complex analysis have played a central role in answering questions in this subject at the interface of harmonic analysis and PDE.
Brief Biography:
Jill Pipher has made a profound impact on mathematics, both through her work in the fields of harmonic analysis and partial differential equations, and though her service to the profession. Pipher is the Elisha Benjamin Andrews Professor at Brown University, where she has served on the faculty since 1989. Before that, she was a L. E. Dickson Instructor and Assistant Professor at the University of Chicago. Her PhD was awarded by UCLA in 1985, and directed by John Garnett.
Pipher is best known for her fundamental contributions to solutions and regularity of partial differential equations in minimally smooth domains. For example, her classic 1995 paper with Verchota, Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators (Ann. of Math.), settled a long-standing conjecture on the solvability of the Dirichlet problem with L2 boundary data on bounded Lipschitz domains.  More recently, with Hofmann, Kenig, and Mayboroda, Pipher introduced innovative new tools to establish solvability of the Dirichlet problem with Lp data for non-symmetric elliptic operators.
Pipher has also done groundbreaking work in cryptography. With her collaborators Hoffstein, and Silverman, Pipher described the first secure and practical public key cryptosystem based on hard lattice problems (NTRU). NTRU appears to be secure against attack by quantum computers, unlike earlier systems including RSA. This work has been influential, spawning an intense new research area of lattice-based cryptosystems. Pipher holds four patents in encryption.
Pipher is the founding director for the Institute for Computational and Experimental Mathematics (ICERM) at Brown. She also served as AWM president 2011-2013.