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The University of New Mexico

Colloquium on "Propagation of randomness under the flow of nonlinear dispersive equations" by Andrea Nahmod (University of Massachusetts, Amherst)

Event Type: 
Colloquium
Speaker: 
Andrea Nahmod (University of Massachusetts, Amherst)
Event Date: 
Thursday, April 15, 2021 -
3:30pm to 4:30pm
Location: 
Zoom Meeting ID: 937 6606 4787
Audience: 
General Public

Event Description: 

Title: Propagation of randomness under the flow of nonlinear dispersive equations.
 
Abstract: The study of partial differential equations (PDEs) with randomness has become an important and influential subject in the last few decades. In this talk we focus on the time dynamics of solutions of nonlinear dispersive equations with random initial data. It is well known that in many situations, randomization improves the behavior of solutions to PDEs: the key underlying difficulty is in understanding how randomness propagates under the flow of nonlinear PDEs. In this context, starting with an overview of J. Bourgain's seminal work on the invariance of Gibbs measures for nonlinear Schrödinger equations we describe new methods that offer deeper insights. We discuss in particular the theory of random tensors, a powerful new framework that we developed with Yu Deng and Haitian Yue, which allows us to unravel the propagation of randomness beyond the linear evolution of random data and probe the underlying random structure that lives on high frequencies/fine scales. This enables us to show the existence and uniqueness of solutions to the NLS in an optimal range relative to the probabilistic scaling. A beautiful feature of the solution we find is its explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients.
 

About the Speaker: Andrea Nahmod received her Licenciatura from the University of Buenos Aires, Argentina and her PhD from Yale University in 1991 under the direction of Ronald Coifman. After postdoctoral positions at McQuarie University at Sydney, Australia, University of Texas at Austin, the Mathematical Sciences Research Institute (MSRI) at Berkeley,  and the Institute for Advanced Studies (IAS) at Princeton, she joined in 1998 the faculty at the University of Massachusetts at Amherst, where she is now a Full Professor. Dr. Nahmod research interests lie at the interface of non-linear Fourier and Harmonic Analysis and the theory of PDE modeling wave propagation phenomena. She has been leading research at the forefront of non-linear PDEs aimed at gaining a quantitative undertsanding of randomness in the dynamical evolution on non-linear waves in various regimes.
In 2014, Dr. Nahmod became a fellow of the American Mathematical Society. The society cited her “contributions to nonlinear Fourier analysis, harmonic analysis, and partial differential equations, as well as service to the mathematical community.” She is also a recipient of a Sargent-Faull Fellowship at Harvard's Radcliffe Institute for Advanced Study and a Simons Fellowship. She has had NSF support since 1999 and she is currently a PI af an 8M Simons MSP collaboration award on Wave Turbulence. Dr. Nahmod has a remarkable record mentoring postdocs, PhD and undergraduate students. She has organized workshops and semester long programs  at various intitutes, including the upcoming ICERM Semester Program on Hamiltonian Methods in Dispersive and Wave Evolution Equations in Fall 2021.

In October 2007, Andrea was Main Lecturer in the 10th NM Analysis Seminar where she taught a minicourse on "Bilinear Operators in Analysis and PDEs". In Spring 2014, Andrea co-organized with Wilfredo Urbina and Cristina Pereyra and "Afternoon in Honor of Cora Sadosky", this even took place right after the 13th NM Analysis Seminar and before an AMS meeting we hosted at UNM.

Event Contact

Contact Name: María Cristina Pereyra

Contact Email: crisp@math.unm.edu