Instructor: Jon Wilkening

Lectures: MWF 9:10-10:00, Room 75 Evans

Office: 1051 Evans

Office Hours: Mon 10:15-11:45, Thurs 2:30-4:00

Prerequisites: Undergraduate Analysis (104), Complex Analysis (185), and Linear Algebra (110)

Required Text:
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I

Recommended Reading:
Gabor Szego, Orthogonal Polynomials
Barry Simon, Orthogonal Polynomials on the Unit Circle
Lloyd Trefethen, Approximation Theory and Approximation Practice
Reed and Simon, Functional Analysis, Vol I
Courant and Hilbert, Methods of Mathematical Physics, vol 1
Coddington and Levinson, Theory of Ordinary Differential Equations
Ivar Stackgold, Green's Functions and Boundary Value Problems
Novikov, Manakov, Pitaevskii, Zakharov, Theory of Solitons, The Inverse Scattering Method

Catalog description: Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green functions. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves.

Syllabus: The course will survey basic theory and practical methods for solving the fundamental problems of mathematical physics. It is intended for graduate students in applied mathematics, physics, engineering or other mathematical sciences. The overall purpose of the course will be to develop non-numerical tools for understanding and approximating solutions of differential equations.

• Hilbert Spaces, Orthogonal Polynomials, Approximation Theory (Chapter 1 + Other Books)
• Distributions, Tempered Distributions, Fourier Transform (Chapters 2-4)
• Linear Operators on a Hilbert Space, Spectrum and Resolvent (Chapters 7,8)
• Spectral Theory for Compact Operators, Green's functions, Regular Sturm-Liouville problems (Other books)
• Self-Adjoint and Unitary Operators, Continuous Spectrum, Singular Sturm-Liouville problems (Chapters 10-12 + Other books)
• KdV and the Inverse Scattering Transform (Time permitting. Otherwise bumped to 224B)

Course Material: I will post handouts, assignments and solutions on bCourses. Please e-mail me if you do not have access to the bCourses page.

Grading: 80% Homework, 20% Class Project (groups of 1-3. Pick something that interests you, relevant to the course, that would be equivalent to 2 assignments in scope for each member of the group)

Homework: 8 assignments + project

Comments: I'll drop the two lowest homework scores, or the project (if you really are not interested in doing one.)