Title: The Schrödinger equation as inspiration of beautiful mathematics.

Abstract: In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

About the Speaker:  Gigliola Staffilani did her undergraduate studies at the University of Bologna in Italy, and her PhD in 1995 at the University of Chicago under the direction of Carlos Kenig. After postdoctoral studies at the Institute for Advanced Study, Stanford University , and Princeton University, Staffilani became a faculty member at Stanford, Princeton, and Brown universities, before moving to the Massachusetts Institute of Technology in 2002 where she now is  the Abby Rockefeller Mauzé Professor of Mathematics. She has twice been a member of the Institute for Advanced Study in Princeton, a member of Mathematical Sciences Research Institute (MSRI), and  a member of the Radcliffe Institute for Advanced Study at Harvard University. She has received several National Science Foundation grants, a large 2019-2023 Simons Collaboration Grant on Wave Turbulence and a Sloan Research Fellowship.  In 2012 she became one of the inaugural fellows of the American Mathematical Society. In 2014 she was inducted into the American Academy of Arts and Sciences. In 2021, she was elected to the National Academy of Sciences.
Staffilani is mainly interested in harmonic analysis and the study of partial differential equations (PDE). She is particularly interested on  the long time behavior of  periodic solutions to nonlinear Schrödinger equations. For these solutions in fact she is investigating questions related to energy transfer and forward cascade,  concepts strictly related to weak turbulence theory. In recent years Staffilani has introduced concepts borrowed from probability theory in oder to establish results, such as well-posedness for certain Schrödinger equations at very low regularity,  that are  “generically” true. Staffilani has trained many graduate students and postdoctoral fellows and  she has earned many teaching awards. She serves the profession in editorial boards of several journals, she is a member of the AMS Executive Board and Council, organizing conferences and workshops, for example this semester she is organizing an ICERM Program: Hamiltonian Methods in Dispersive Wave Evolution.