Professor Andrea R. Nahmod of Mathematics and Statistics has been awarded a Simons Foundation Fellowship in Mathematics by the Simons Foundation.
With the award, Nahmod will spend her 2013-14 sabbatical year as a visiting professor at the Massachusetts Institute of Technology, where she will pursue ongoing research with MIT’s Gigliola Staffilani on deterministic and nondeterministic aspects of nonlinear partial differential equations (PDEs).
Nahmod’s research lies at the overlap of nonlinear Fourier analysis, harmonic analysis, and nonlinear partial differential equations, integrating into it tools from geometry, gauge theory, and probability. Her research builds on and adds to the extensive progress made during the last two decades in settling fundamental questions on the existence of solutions to nonlinear dispersive and wave equations, their long time behavior, and the formation of singularities. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena for which sophisticated tools from nonlinear Fourier analysis, geometry, and analytic number theory have played a crucial role. Yet, there remain some fundamental obstacles. A natural approach to overcome them is to consider evolution equations from a nondeterministic point of view and incorporate, in the analysis, tools from probability as well. The convergence of deterministic and probabilistic approaches, together with the extraordinary maturity of the field at this point, make this the perfect time for attacking some of the most challenging and fascinating questions in this area, says Nahmod.
Wave phenomena in physics are mathematically modeled using PDEs. Nonlinear wave models arise in many fields including quantum mechanics, ferromagnetism, vibrating systems, semiconductors, and optical fibers. Dispersive PDEs model certain wave-propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves while conserving energy or mass. The best-known dispersive PDEs are the nonlinear Schrödinger equations governing the motion of quantum particles.
The role of mathematical analysis, and the focus of Nahmod’s research, is to understand the behavior of the solutions of these dispersive PDEs and to provide the tools to extract their quantitative and qualitative information. The synergy of Fourier analysis, probability, geometry, and analytic number theory provides a well-adapted and powerful toolbox to study the nonlinear effects that allow waves to interact to produce new modified propagation patterns.
Simons Fellows are chosen on the basis of their research accomplishments during the five years prior to application and on the basis of the potential scientific impact of the fellowship.
Photo by Tony Rinaldo