Advanced Real Analysis

Math 212a / 10:30-11:45 Tu Th / Science Center room 507
Harvard University - Fall 2019
Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Required Texts
  • Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991
Recommended Texts
  • Berberian, Lectures on Functional Analysis. Springer-Verlag, 1974
  • Reed and Simon, Functional Analysis I. Academic Press, 1980.
  • Riesz-Nagy, Functional Analysis. Dover reprint, 1990.
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. Math 113 and 114 are prerequisites. This includes complex function theory, Lebesgue measure and integral, general topology, and the theory of linear operators between Banach spaces.

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, Property T, expanding graphs, and further topics
Homework. Collaboration is encouraged on homework. Late homework is not accepted, but the lowest homework grade will be dropped.

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.

Classroom policies. Please do not use cellphones, laptops or keyboards in class and do not record the lectures. The use of tablet computers with a stylus for note-taking is permissible.

Course home page: http://math.harvard.edu/~ctm/math212a