Wilhelm Schlag's Home Page

Wilhelm Schlag

Department of Mathematics
Yale University
10 Hillhouse
New Haven, CT 06520

Office phone: (203) 432-4190
Fax: (203) 432-7316
Email: wilhelm dot schlag dot yale dot edu


Editorial boards

Recent papers

  1. Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle
    with Rui Han, Marius Lemm, preprint 2018.

  2. A higher dimensional Bourgain-Dyatlov fractal uncertainty principle
    with Rui Han, preprint 2018.

  3. Effective multi-scale approach to the Schrödinger cocycle over a skew shift base
    with Rui Han, Marius Lemm, preprint 2018.

  4. Intertwining wave operators, Fourier restriction, and Wiener theorems
    W. Schlag, preprint 2018.

  5. Long time dynamics for weakly damped nonlinear Klein-Gordon equations
    with Nicolas Burq, Genevieve Raugel, preprint 2018.

  6. On the Spectrum of Multi-Frequency Quasiperiodic Schrödinger Operators with Large Coupling
    with Michael Goldstein, Mircea Voda, preprint 2017.

  7. Global center stable manifold for the defocusing energy critical wave equation with potential
    with H. Jia, B. Liu, G. Xu, preprint 2017.

  8. Structure formulas for wave operators under a small scaling invariant condition
    with M. Beceanu, preprint 2017.

  9. Structure formulas for wave operators
    with M. Beceanu, preprint 2016.

  10. On localization and the spectrum of multi-frequency quasi-periodic operators
    with M. Goldstein, M. Voda, preprint 2016.

  11. Cartan Covers and Doubling Bernstein Type Inequalities on Analytic Subsets of C^2
    with M. Goldstein, M. Voda, preprint 2016.

  12. Linear stability of the Skyrmion
    with M. Creek, R. Donninger, S. Snelson, preprint 2016.

  13. The Weber equation as normal form with applications to top of the barrier scattering
    with R. Costin, H. Park, preprint 2015.

  14. Generic and non-generic behavior of solutions to the defocusing energy critical wave equation with potential in the radial case
    with H. Jia, B. Liu, G. Xu, preprint 2015

  15. Long time dynamics for damped Klein-Gordon equations
    with N. Burq, G. Raugel, preprint 2015

  16. Homogeneity of the spectrum for quasi-periodic Schroedinger operators
    with D. Damanik, M. Goldstein, M. Voda, preprint 2015

  17. Stable soliton resolution for exterior wave maps in all equivariance classes
    with C. Kenig, A. Lawrie, B. Liu, preprint 2014

  18. Channels of energy for the linear radial wave equation
    with C. Kenig, A. Lawrie, B. Liu, preprint 2014

  19. Semilinear wave equations, ICM proceedings, Seoul Korea August 2014,

  20. Large global solutions for energy supercritical nonlinear wave equations on R^{3+1}
    with Joachim Krieger, preprint 2014

  21. Profiles for the radial focusing 4d energy-critical wave equation
    with R. Cote, C. Kenig, A. Lawrie, preprint 2014

  22. Center-stable manifold of the ground state in the energy space for the critical wave equation
    with Joachim Krieger and Kenji Nakanishi, preprint 2013

  23. Relaxation of wave maps exterior to a ball to harmonic maps for all data
    with Carlos Kenig, Andrew Lawrie, preprint 2013

  24. Exotic blowup solutions for the u^5 focusing wave equation in R^3
    with R. Donninger, M. Huang, J. Krieger, preprint 2012

  25. Full range of blow up exponents for the quintic wave equation in three dimensions
    with Joachim Krieger, preprint 2012

  26. The method of concentration compactness and dispersive Hamiltonian evolution equations.
    Proceedings from the 2012 ICMP at Aalborg, Denmark.

  27. Regularity and convergence rates for the Lyapunov exponents of linear co-cycles
    preprint 2012

  28. Characterization of large energy solutions of the equivariant wave map problem: II
    with Raphael Cote, Carlos Kenig, Andrew Lawrie, preprint 2012

  29. Characterization of large energy solutions of the equivariant wave map problem: I
    with Raphael Cote, Carlos Kenig, Andrew Lawrie, preprint 2012

  30. Energy partition for the linear radial wave equation
    with Raphael Cote, Carlos Kenig, preprint 2012

  31. Threshold phenomenon for the quintic wave equation in three dimensions
    with Joachim Krieger, Kenji Nakanishi, preprint 2012


Slides from talks

  1. ICM talk

  2. Introduction, Lecture 1, Lecture 2, Lecture 3, Lecture 4
    Anderson localization, small divisors, and subharmonic functions
    Slides for the September 15-21 2013 meeting at Maiori, Italy, and the minicourse in Vienna, Austria, 09/25-27, 2013.
    Many thanks to Michael Goldstein and Silvius Klein for helpful suggestions
    which improved the presentation.

  3. Concentration compactness and dispersive Hamiltonian equations
    Slides from the ICMP at Aalborg, Denmark, August, and IMA, Minneapolis, September 2012

  4. Decay of waves on a curved background
    These are an extended form of slides used in various talks describing the dispersive decay of waves
    on curved backgrounds with trapping. Examples are surfaces of revolution which are asymptotically conic, but also the
    Schwarzschild black hole background.

  5. Invariant manifolds for dispersive Hamiltonian evolution equations
    These slides describe the global dynamics of solutions with energies near that of unstable ground state solitons for dispersive
    certain Hamiltonian evolution equations.

  6. Invariant manifolds and dispersive Hamiltonian evolution equations
    This is an overview talk, delivered at the joint AMS/MAA meeting, Boston, January 5, 2012.


  • Concentration compactness for critical wave maps, with Joachim Krieger.
    This is our proof of global existence for large data energy critical wave maps into
    the hyperbolic plane based on the profile decomposition.
    Has appeared in the series "Monographs" of the EMS Publishing house: EMS

  • Invariant Manifolds and dispersive Hamiltonian Evolution Equations, with Kenji Nakanishi.
    This is based on a course taught at ETH Zürich in the fall of 2010.
    Many thanks to Martin Sack at ETH for his help in preparing these lecture notes.
    Has appeared in ''Zürich lectures in Advanced Mathematics'' of the EMS Publishing House.
    The link to the publisher page is here EMS. In North America, the book can be ordered via the AMS, see ORDER.

  • Classical and Multilinear Harmonic Analysis, with Camil Muscalu.
    The first volume of this book, which covers classical harmonic analysis, is a much
    revised and expanded version of my old harmonic analysis notes, see below.
    There is a completely new multilinear part as a second volume which covers Carleson's
    theorem, the bilinear Hilbert transform and much more. To appear early 2013
    with Cambridge University Press. See here for Vol 1 and Vol 2


Meetings and Conferences

  • PDE Conference in Ascona
  • This meeting brought together experts in completely integrable equations and KAM theory with those using more
    harmonic analysis oriented methods. It took place in Ascona, Switzerland, July 1-6, 2012. It was organized by
    Dario Bambusi, Thomas Kappeler, Joachim Krieger and myself.

  • Joint AMS/MAA meeting, Boston
  • Together with Eugene Wayne from Boston University, I organized a special session on
    "Stability Analysis for Infinite Dimensional Hamiltonian Systems" at the Joint Mathematics Meeting that was held
    Jan. 4-5, 2012 in Boston.
    More information can be found here: AMS

Numerical Data

  • Numerical results for radial nonlinear Klein-Gordon in three dimensions
    Under this link the reader will find most of the numerical data as well as C-codes
    which were produced together with Roland Donninger in 2010 using a MacPro 8-core work station.
    These data can only be understood in conjunction with the paper 'Numerical study of the
    blowup/global existence dichotomy etc.' which can be found above. Although the authors have also
    conducted computations on the NICS CRAY XT5 supercomputer which are described in this paper,
    we have only deposited the MacPro material here as the CRAY uses MPI and is not so easily portable.
    For the most part, the data are accompanied by readme files which offer some information about the codes.
    The directory linked here is simply a copy of a (somewhat cleaned up) working directory which was used in the process of
    producing the data. The executables were erased, as well as the .pdf files of the pictures which are simply too large.
    All pictures are in .png format, and they were produced from the data files (.dat) via gnuplot.