**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.)