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Bulletin of the American Mathematical Society

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The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability


Author: Andrea R. Nahmod
Journal: Bull. Amer. Math. Soc. 53 (2016), 57-91
MSC (2010): Primary 42-XX, 35-XX
DOI: https://doi.org/10.1090/bull/1516
Published electronically: September 1, 2015
MathSciNet review: 3403081
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Abstract: The field of nonlinear dispersive and wave equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry, analytic number theory and most recently probability, into the existing functional analytic methods. In these lectures we concentrate on the semilinear Schrödinger equation defined on tori and discuss the most important developments in the analysis of these equations. In particular, we discuss in some detail recent work by J. Bourgain and C. Demeter proving the $ \ell ^2$ decoupling conjecture and as a consequence the full range of Strichartz estimates on either rational or irrational tori, thus settling an important earlier conjecture by Bourgain.


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Additional Information

Andrea R. Nahmod
Affiliation: Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 North Pleasant Street, Lederle GRT, Amherst, Massachusetts 01003
Email: nahmod@math.umass.edu

DOI: https://doi.org/10.1090/bull/1516
Received by editor(s): May 18, 2015
Published electronically: September 1, 2015
Additional Notes: The author gratefully acknowledges support from NSF grants DMS 1201443 and DMS 1463714.
Article copyright: © Copyright 2015 American Mathematical Society