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Presidential Lectures

How Do Waves Propagate Randomness?

  • Speaker
  • Andrea R. Nahmod Professor of Mathematics Department of Mathematics University of Massachusetts Amherst, Professor, Department of Mathematics and Statistics Within the international research community, Andrea R. Nahmod is one of the most visible members of UMass Amherst’s Department of Mathematics and Statistics. A native of Argentina, she works in nonlinear analysis and partial differential equations, a subfield of pure mathematics, but has also collaborated with applied mathematicians. “I’m an analyst,” Nahmod says. “I study how to decompose objects in forms we can understand and that give us information about their most relevant features, their structure and patterns.” Using harmonic and nonlinear Fourier analysis, Nahmod applies these decomposition techniques to problems in the material world in order to find solutions and to understand their behavior. Nahmod was named Professeur Invité at Université Paris-Sud 11, Orsay, and was a visiting scholar at the Courant Institute, New York University. She was awarded a Radcliffe-Sargent Faull Fellowship at Harvard University and has been twice selected a member of the Institute for Advanced Study at Princeton. Nahmod's research has been continuously funded by the National Science Foundation. She delivered an Invited Plenary Address at the American Mathematical Society in 2006 and was keynote speaker at the Tenth New Mexico Analysis conference in 2007, the 16th Riviere-Fabes Symposium on Analysis and PDE in 2013, and the 33th SEARCDE Conference in 2014. She frequently speaks at major research conferences in the U.S. and overseas, including the Oberwolfach Institute of Mathematics in Germany in 2013, the NSF-CBMS Regional Research Conference in 2013, the Banff International Research Station in Canada in 2014, and the International Conference on Recent Progress and Future Directions at the University of Chicago in 2014. Education: BS, University of Buenos Aires, 1985; PhD, Yale University, 1991.Andrea Nahmod, Ph.D.Professor of Mathematics, University of Massachusetts Amherst
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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.

About the Speaker

Andrea R. Nahmod Professor of Mathematics Department of Mathematics University of Massachusetts Amherst, Professor, Department of Mathematics and Statistics Within the international research community, Andrea R. Nahmod is one of the most visible members of UMass Amherst’s Department of Mathematics and Statistics. A native of Argentina, she works in nonlinear analysis and partial differential equations, a subfield of pure mathematics, but has also collaborated with applied mathematicians. “I’m an analyst,” Nahmod says. “I study how to decompose objects in forms we can understand and that give us information about their most relevant features, their structure and patterns.” Using harmonic and nonlinear Fourier analysis, Nahmod applies these decomposition techniques to problems in the material world in order to find solutions and to understand their behavior. Nahmod was named Professeur Invité at Université Paris-Sud 11, Orsay, and was a visiting scholar at the Courant Institute, New York University. She was awarded a Radcliffe-Sargent Faull Fellowship at Harvard University and has been twice selected a member of the Institute for Advanced Study at Princeton. Nahmod's research has been continuously funded by the National Science Foundation. She delivered an Invited Plenary Address at the American Mathematical Society in 2006 and was keynote speaker at the Tenth New Mexico Analysis conference in 2007, the 16th Riviere-Fabes Symposium on Analysis and PDE in 2013, and the 33th SEARCDE Conference in 2014. She frequently speaks at major research conferences in the U.S. and overseas, including the Oberwolfach Institute of Mathematics in Germany in 2013, the NSF-CBMS Regional Research Conference in 2013, the Banff International Research Station in Canada in 2014, and the International Conference on Recent Progress and Future Directions at the University of Chicago in 2014. Education: BS, University of Buenos Aires, 1985; PhD, Yale University, 1991.

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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