Waves are everywhere in nature. They arise in quantum mechanics, fiber optics, ferromagnetism, the atmosphere, water and many other models. Such wave phenomena are never too smooth or simple — the byproduct of nonlinear interactions. Understanding and describing the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and having a precise description of how the inherent randomness built in these models propagates is fundamental to accurately predicting wave phenomena when studying the natural world.

In this talk, Andrea Nahmod will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. She will continue by explaining the fundamental paradigm shift that arises from the “correct” scaling in this context and how it opened the door to unveiling the random structures of nonlinear waves that live on high frequencies and fine scales. She will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated with nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and the hyperbolic Φ^4_3 model in the context of constructive quantum field theory. Finally, she will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.

Nahmod is a professor of mathematics at the University of Massachusetts, Amherst. She is best known for her work on nonlinear partial differential equations and nonlinear Fourier analysis. After receiving her Ph.D. from Yale University, she held research and faculty positions at Macquarie University, Sydney, the University of Texas at Austin, the Mathematical Sciences Research Institute in Berkeley, California and the Institute for Advanced Study in, Princeton, New Jersey, before joining UMass Amherst. Nahmod was the Sargent Faull Fellow at the Radcliffe Institute for Advanced Study at Harvard and a Simons Foundation Fellow. She held Simons Professorships at MSRI and at the Centre de Recherches Mathématiques, Montreal. At UMass Amherst, Nahmod received the Award for Outstanding Accomplishments in Research and Creative Activity and a Samuel E. Conti Fellowship, two of the top honors. Nahmod is a fellow of the American Mathematical Society. The National Science Foundation has continuously funded her research, and she is a principal investigator in the Simons Collaboration on Wave Turbulence.

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SCHEDULE

Doors open: 5:30 p.m. (No entrance before 5:30 p.m.)

Lecture: 6:00 p.m. – 7:00 p.m. (Admittance closes at 6:20 p.m.)

The Gerald D. Fischbach Auditorium complies with the Americans with Disabilities Act and offers accessible seating to visitors with special access needs.

The theme of the 2023 Simons Foundation Presidential Lecture series in mathematics and computer science is Waves. The equations that describe waves are crucial tools for understanding everything from how sound propagates to the behavior of quantum particles. This set of talks will explore the richness of the wave equation and its nonlinear extensions in general relativity, particle physics, hydrodynamics, optics, quantum mechanics and electrodynamics.

HOW DO WAVES PROPAGATE RANDOMNESS?