Class schedule
Note: Click on the date to get Ibrahim's class notes. He has also written some supplementray notes:Wed 1/11 | Idea of distributions. Topological vector spaces. |
Fri 1/13 | Topological vector spaces. Hausdorff property. |
Wed 1/18 | Locally convex spaces. Seminorms. Fréchet spaces. |
Fri 1/20 | LF spaces. |
Wed 1/25 | Distributions. Radon measures. |
Fri 1/27 | Subspaces of distributions. Basic operations on distributions. |
Wed 2/1 | Sheaf structure of distributions. |
Fri 2/3 | Local structure of distributions. |
Wed 2/8 | Convolution. |
Fri 2/10 | Constant coefficient operators. Fundamental solutions. Hypoellipticity. |
Wed 2/15 | Schwartz theorem. Laurent expansion. Analytic hypoellipticity. |
Fri 2/17 | Fourier transform. Liouville's theorem. |
2/20–2/24 | Study break |
Mon 2/27 | Hörmander's characterization of hypoelliptic polynomials. |
Fri 3/2 | Problems in half-space. Cauchy problem. Petrowsky well-posedness. |
Wed 3/7 | Boundary value problems. Lopatinsky-Shapiro condition. |
Fri 3/9 | Strongly hyperbolic and p-parabolic systems. |
Wed 3/14 | Strong hyperbolicity. Inhomogeneous Cauchy problem. |
Fri 3/16 | Well-posedness of a general class of Cauchy problems. Parabolicity. |
Wed 3/21 | Semilinear evolution equations. |
Fri 3/23 | Multiplication in Sobolev spaces. Derivative nonlinearities. |
Wed 3/28 | Elliptic boundary value problems. Gårding inequality. |
Fri 3/30 | Gårding inequality proof. Dirichlet problem. Lax-Milgram lemma. |
Wed 4/4 | Friedrichs inequality. Rellich-Kondrashov compactness. |
Fri 4/6 | Good Friday |
Wed 4/11 | L2-regularity theory. |
Fri 4/13 | Spectral theory. Semigroups. |
Assignments
Final project
Date | ||
4/16 | Morgane Henry | Wave maps |
4/16 | Ibrahim Al Balushi | Nonlinear diffusion |
4/16 | Olga Yakovlenko | Introduction to the Ricci flow |
4/30 | Yang Guo | Einstein equations |
4/30 | Sebastien Picard | Introduction to the Yang-Mills equations |
4/30 | Spencer Frei | DeGiorgi-Nash-Moser regularity theory |
4/30 | Mario Palasciano | A nonlocal aggregation model |
4/30 | Olivier Mercier | Mean curvature flow |
4/30 | Joshua Lackman | Pseudodifferential operators |
4/30 | Andrew MacDougall | Some topics in semiclassical analysis |
The final project consists of the student studying an advanced topic, typing up expository notes, and presenting it in class. Here are some ideas for the project:
Weekly seminars
PDE questions from previous qualifying exams for download.Date | |||
1/16 | Baire's theorem and consequences | Rudin Ch2, Tao | Gantumur |
1/23 | Hahn-Banach theorem | Rudin §3.1-3.7, Tao | Spencer |
1/30, 2/6 | Closed range theorem | Rudin §4.1-4.15 | Ibrahim |
2/13, 2/29 | Fredholm operators | Rudin §4.16-4.25, McLean 2.14-2.17, Tao | Mario |
3/5 | Banach-Alaoglu theorem | Rudin §3.8-3.18, Tao | Andrew |
3/12, 3/19 | Spectral theorem | Rudin Ch13, Jaksic | Sébastien |
3/26 | Hille-Yosida theorem | Rudin §13.34-13.37 | Yang |
4/2 | Navier-Stokes equations | Tao, Clay, Lei-Lin | Gantumur |
References
Topics to be covered
Instructor
Dr. Gantumur TsogtgerelOffice: Burnside Hall 1123. Phone: (514) 398-2510.
Email: gantumur -at- math.mcgill.ca.
Office hours: Just drop by or make an appointment
Online resources
PDE Lecture notes by Bruce Driver (UCSD)Xinwei Yu's page (Check the Intermediate PDE Math 527 pages)
John Hunter's teaching page at UC Davis (218B is PDE)
Textbook by Ralph Showalter on Hilbert space methods
Lecture notes by Georg Prokert on elliptic equations