Real Analysis

Advanced Real Analysis

Math 212b / Tu Th 10-11:30 / 216 SC
Harvard University - Spring 2006

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

Required Texts

Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991
Berberian, Lectures on Functional Analysis. Springer-Verlag, 1974 (unfortunately, out of print)

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. Directed to graduate students. Lebesgue measure and integral, general topology, and theory of linear operators between Banach spaces will be assumed.

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
- Fluid flow in the plane
- 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, Property T, expanding graphs, and further topics

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 exams. Collaboration is encouraged on homework. Exam work should be based only on course materials and done without the help of others.

2006 Calendar.

2 Feb (Th)	First class.
28, 30 Mar (Tu,Th)	No class - spring break.
4 May (Th)	Last class.

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