Pedro Nunes' Lecture 2025

Pedro Nunes Lecture 2025

Michael Christ

Michael Christ (born 7 June 1955) is an American mathematician and Emeritus Professor at the University of California, Berkeley. 

His research has Fourier analysis at its core and encompasses partial differential equations, complex analysis in several variables, and topics in mathematical physics. Among his many outstanding research contributions, we highlight the definitive analysis of global regularity of solutions of $\del$-bar problems on pseudoconvex domains, the proof (with Colliander and Tao) of the ill-posedness of low-regularity solutions of the nonlinear Schrödinger equations, and the characterization (with Kiselev) of the absolutely continuous spectrum and generalized eigenfunctions for second-order ODE with potentials on the real line.

Michael has been an invited lecturer twice at the International Congress of Mathematicians, first in Kyoto in 1990 and then in Berlin in 1998. He has received numerous honors and awards, including an NSF Presidential Young Investigator Award and a Sloan Fellowship, the Bergman Prize from the AMS, and a Miller Research Professorship. He was elected to the American Academy of Arts and Sciences in 2007.

7th of May 2025

14:00, Anfiteatro Abreu Faro, Instituto Superior Técnico, Lisbon

9th of May 2025

14:30, Sala 0.07, Edifício das Matemáticas, Faculdade de Ciências da Universidade do Porto

Oscillatory Multilinear Inequalities

Abstract: Integrals involving rapidly oscillatory factors abound. Quintessential examples include the Fourier transform, integral operators employed in the analysis of wave phenomena, and exponential sums. The concept of stationary phase is a central element of the theory.

The Fourier transform can be viewed as a linear operator, or a bilinear form. We focus on multilinear operators, which are less canonical than the Fourier transform and are open to structural questions reflecting their diversity. 

We begin with certain multilinear inequalities which do not involve cancellation, generalizing convolution. We then bring oscillation into the picture, and finally discuss inequalities for multilinear forms 
in which the oscillation is implicit in the structure of the inequalities, rather than appearing explicitly in the integrals. The notion of stationary phase is reborn in mutated form.

Along the way we will see connections to web geometry, Hilbert's tenth problem, additive combinatorics, convex geometry, Ramsey theory, computer science, weak continuity, and ergodic theory.