Walter Craig
November 28, 1953 - January 18, 2019

Mathematics 4FT/6FT: Topics in Partial Differential Equations:
Fourier integral and pseudodifferential methods

Course description and objectives The Fourier transform has many broad applications to wide range of topics, not the least being to partial differential equations. In fact, from one point of view the Fourier transform provides a system of coordinates in Hilbert space in which all constant coefficient differential operators are diagonal. The first part of this course will describe settings in which the Fourier transform and the Fourier inversion theorem apply. This includes the standard Hilbert space L²(Rⁿ), Sobolev spaces, Schwartz Class, and the spaces of distributions and tempered distributions. The second part of this course focuses on pseudodifferential operators and their use in analysis. This class of operators includes differential operators with variable coefficients, fractional derivatives, and otherwise operators that allow one to microlocalize analysis; that is, to localize analytic considerations simultaneously in position space and in Fourier space.

The course material wll be presented at an advanced undergraduate/masters student level. Despite the apparently advanced and technical nature of the topic, it is actually quite intuitive, a fact that will be emphasized in the course.

The principal goals of the course are: (1) to understand properties of solutions to partial differential equations given through Fourier synthesis, including solutions with classical regularity properties as well as distributional solutions. And (2) to introduce students to the calculus of pseudodifferential operators, opening up the possibilities of a local and microlocal analysis of distributions and their singularities.

Textbooks
1. A Course on Partial Differential Equations by W. Craig, published in McMaster courseware.
2. Pseudodifferential Operators by Michael Taylor, Princeton University Press.
The textbooks are required. You will be reading from it regularly, and homework assignments will include selected problems from the two books.

Syllabus and Course Outline: Topics in Partial Differential Equations

Mathematics 3FF: Partial Differential Equations

Class: M & Th 15:30 - 16:20 and Tu 16:30 - 17:20 in ABB 271
Office Hours: F 10:30 - 12:00 or by appointment

Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many of the laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. The course Math 3FF addresses the principal partial differential equations that arise in physical contexts, where the course material will present explicit methods for solving them. In the course of this analysis we will deduce many of the main features of these solutions, including finite propagation speed and conservation of energy for wave equations, maximum principles for heat flow, smoothness of solutions to Laplace's equations, and other mathematical aspects of familiar and less familiar laws of physics.

Course Syllabus and Course Outline: Partial Differential Equations

Partial differential equations arise in a physical context in descriptions of phenomena such as fields and forces. In mathematics partial differential equations are used to describe geometrical objects as well as concepts in analysis. The version of the course Mathematics 4FT of this semester will concentrate on three sets of techniques that are closely associated with differential equations. these are (1) Hilbert space methods, including in particular the Fourier transform, (2) the method of characteristics, and (3) majorant methods, which involve classes of holomorphic (analytic) functions. The course material will include both linear and nonlinear partial differential equations, it will cover classical as well as modern techniques, and it will make an emphasis on the applications of the theory to problems in physics and geometry.

Course Syllabus and Course Outline: Topics in Partial Differential Equations

Research interests: partial differential equations and dynamical systems

Journal editorships:

  AMS, Mathematical Surveys and Monographs;  February 1 2018 - January 31 2022.

  Communications in Contemporary Mathematics;  2006 - present.

  Complex Analysis and its Synergies;  2014 - present.

  Electronic Research  Announcements in Mathematical Sciences; (ERA-MS) February 15 2007 - February 1 2019.

  Fields Institute, Editorial Board;   2001 - present.

  Journal of Dynamics and Differential Equations;  2006 - present.

  Mathematical Reports of the Royal Society of Canada;   2008 - present.

  Nonlinear Differential Equations and Applications;  2008 - present.

Journal editorships (past):

   London Mathematical Society Monographs, 2004 - 2008.            

   Philosophical Transactions of the Royal Society - A:   January 2009 - December 2011.

   Proceedings of the AMS   Feb. 2005 - Jan. 2013.

   Mathematical Physics Electronic Journal,   2002 - 2015

   AMS, Graduate Studies in Mathematics   Feb. 2002 - Jan. 2008.

   Canadian Mathematical Society, Journal and Bulletin   2002 - 2007.

  Proceedings of the Royal Society - A:   2002 - 2008. (Proc. R. Soc A)

   SIAM: Mathematical Analysis   1998 - 2004.

Curriculum Vitae: cv.pdf file

Selected recent papers:

[107] ''On the Hamiltonian for water waves'',
Proceedings of the RIMS Symposium on Mathematical Analysis of Fluid and Gas Dynamics,
(2016) Research Inst. Math. Sciences - Kyoto, Kokyuroku.

[104] ''Mapping properties of normal forms transformations for water waves''
(with C. Sulem), Bollettino dell'Unione Matematica Italiana 9 (2016),
DOI: 10.1007/s40574-016-0078-9, pp. 289-318.

[102] ''Standing waves in near-parallel vortex filaments'',
(with C. Garcia - Azpeitia and C.-R. Yang), Commun. Math. Phys. 350 (2017) pp. 175-203.
DOI:10.1007/s00220-016-2781-x.

[101] ''Normal forms transformations for capillary-gravity water waves'',
(with C. Sulem), Fields Institute Communications 75 (2015),
eds P. Guyenne, D. Nicholls and C. Sulem,
DOI: 10.1007/978-1-4939-2950-4.

[100] ''On the initial value problem for the wave equation in Friedmann - Robertson - Walker space-times'',
(with B. Abbasi), (2014) Proceedings Royal Society - A 470 20140361.

[97] ''Birkhoff normal form for the nonlinear Schroedinger equation'',
(with A. Selvitella and Y. Wang), (2013)
Rendiconti Accad. Lincei (9) Mat. Appl. 24215-228.

[95] ''The surface signature of internal waves'' (with P. Guyenne and C. Sulem),
Journal of Fluid Mechanics 710 (2012) pp. 277-303.

[93] ''Water waves over a rough bottom in the shallow water regime'' (with D. Lannes and C. Sulem),
Annales IHP - Analyse Nonlineaire 29, (2012) 233-259. 10.1016/j.anihpc.2011.10.004.

[86] ''On the size of the Navier - Stokes singular set'' (with M. Arnold),
DCDS 23 no. 8 (2010).

[79] ``Bounds on Kolmogorov spectrum for the Navier - Stokes equations'',
(with A. Biryuk), Physica D (2012). pdf-file

[78] ``Towards a new proof of Anderson localization'', (with R. Brandenberger),
archived on ArXiv-0805.4217 hep-th (2008). pdf-file

[77] ``Long wave expansions for water waves over random topography'',
(with A. deBouard, O. Diaz-Espinosa, P. Guyenne and C. Sulem)
submitted to Nonlinearity (2008). pdf-file

[75] Hamiltonian dynamical systems and applications (W. Craig, editor),
Proceedings of the Advanced Study Institute on Hamiltonian Dynamical Systems and Applications,
NATO Science for Peace and Security Series B: Springer - Verlag, (2008) XVI, 441~pp.

[76] ``Transformation theory of Hamiltonian PDE and the problem of water waves'',
Proceedings of the Advanced Study Institute on Hamiltonian Dynamical Systems and Applications,
NATO Science for Peace and Security Series B: Springer - Verlag, (2008), pp. 67-83. pdf-file

[72] Electron screening in nanostructures,
(with A. Achoyan, S. Petrosian, H. E. Ruda and A. Shik)
Journal Appl. Phys. 101 104308 (2007), and
Virtual Journal of Nanoscale Science & Technology}, June 4, 2007.

[71] Hamiltonian formulation and long wave models for
internal waves', (with P. Guyenne and H. Kalisch),
Proceedings of OMAE07, 26th International Conference on Offshore
Mechanics and Artic Engineering (2007).

[70] Construction of suitable weak solutions of the Navier - Stokes equations, pdf-file
(with A. Biryuk and S. Ibrahim), Contemp. Math 429 (2006), AMS.

[69] Mathematical aspects of surface water waves, pdf-file
Russian Math. Surveys (2007), in a collection of articles
written by speakers at the Workshop on mathematical hydrodynamics,
Steklov Institute Moscow June 2006.

[68] Surface water waves and tsunamis, pdf-file
J. Dynamics and Differential Equations 18 (2006).

[67] Solitary water wave interactions,
ICMS Workshop on Rogue Waves, December 15, 2005. pdf-file

[66] On the regularity of the Neumann problem for free
surfaces with surface tension, (with A.-M. Matei),
Proceedings AMS 135 (2006).

[65] Stable three-dimensional waves of nearly permanent form
on deep water, (with D. Henderson, M. Oscamou and H. Segur),
Mathematics and Computers in Simulation 74 (2007).

[64] Strong solutions of the Boltzmann equation in
one spatial dimension, (with A. Biryuk and V. Panferov),
C. R. Acad. Sci. Paris  - Mathématiques 342 (2006).

[63] ``Solitary wave interactions'',
(with P. Guyenne, J. Hammack, D. Henderson and C. Sulem),
Physics of Fluids 18 (2006). pdf-file

[62] ``The mathematics of thermal diffusion shocks'',
(with V. Gusev, R. LiVoti, S. Danworaphong and G. Diebold),
 Physical Review E (2005). pdf-file

[61] ``Invariant tori for Hamiltonian PDE'',
Nonlinear Dynamics and Evolution Equations,
editors: Hermann Brunner, Xiao-Qiang Zhao and Xingfu Zou,
Fields Institute Communications, AMS (2005). pdf-file

[60] ``Hamiltonian long wave expansions for free surfaces and interfaces'',
(with P. Guyenne and H. Kalisch), CPAM (2005). pdf-file

[59] ``Thermal Diffusion Shock Waves'',
(with S. Danworaphong, V. Gusev and G. Diebold)
Phys. Rev. Letters 94, p. 095901 (2005). pdf-file

[58] ``Hamiltonian long wave expansions for water waves  over a rough bottom'',
(with P. Guyenne, D. Nicholls and C. Sulem), Proc. Royal Society A 461, (2005), pp. 1 - 35. pdf-file

[39] ``Problèmes de petits diviseurs dans les équations aux dérivées partielles'',
Panaromas et Synthèses 9, Société Mathématiques de France (2000).