Real Analysis

Advanced Real Analysis

Math 212b / Tu Th 10-11:30 / Science Center 507
Harvard University - Spring 1999

Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Course Assistant: Spiro Karigiannis (spiro@math.harvard.edu)

Required Texts

Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991
Berberian, Lectures on Functional Analysis. Springer-Verlag, 1974

Recommended Texts

Riesz-Nagy, Functional Analysis. Dover reprint, 1990.
Reed and Simon, Functional Analysis I. Academic Press, 1980.

Related Literature

Stein, Singular Integrals and Differentiability Properties of Functions.
Princeton University Press, 1970.
de la Harpe et Valette, La Propriété (T) de Kazhdan pour les Groupes Localement Compacts.
Astérisque 175, 1989; distributed by the American Mathematical Society.

Prerequisites. Intended for advanced graduate students.

Topics. This course will present several topics, from elliptic partial differential equations to ergodic theory, with spectral theory and unitary representations as an underlying theme. Possible topics include:

Distributions and Fourier Analysis
- Test functions, convolutions
- Distributions and duality
- Fourier transform and Sobolev spaces
- Singular integral operators
- Elliptic equations
- Prime number theorem
Operator algebras
- Banach algebras and C* algebras
- The Gelfand representation
- Functional calculus
- Positive operators
- Self-adjoint operators
- Unitary operators and representations
Ergodic Theory
- The Birkhoff ergodic theorem
- Geodesic flow on hyperbolic surfaces
- Hopf's proof of ergodicity and mixing
- Kazhdan's property T for SL_3(R)
- Expanding graphs

Grades. Graduate students who have passed their quals are excused from a grade for this course. Grades for other students will be based on homework and a final paper.

Calendar.

4 Feb (Th)	First class.
30 Mar, 1 Apr (Tu,Th)	No class - spring break.
6 May (Th)	Last class. Final papers due.

Course home page: http://math.harvard.edu/~ctm/math212b