Differential Analysis

Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1 hour / session

Prerequisite

Fourier Analysis - Theory and Applications (18.103)

Text

Rudin, W. Real and Complex Analysis. 3rd ed. New York, NY: McGraw-Hill, 1987. ISBN: 0070542341.

Description

The course materials were presented according to the following breakdown of lectures:

• Five lectures on measure and integration - Leading to Riesz representation theorem
• Seven lectures on distributions and Fourier transform - Including Schwart space, Sobolev spaces and Sobolev embedding
• Seven lectures on differential operators with constant coefficients, fundamental solutions and hypoellipticity
• Four lectures on operators, trace class, Hilbert-Schmid

Projects

To pass the course each student was required to carry out one of the projects which were described in the second week of classes. The following projects were suggested to the students:

1. Radon-Nikodym theorem
2. Kuiper’s theorem: The group of unitary operators on a (separable infinite dimensional) Hilbert space is contractible
3. Seeley’s extension theorem
4. Gibb’s phenomenon
5. Surjectivity of any non-trivial constant coefficient differential operator, P: S’ (Rⁿ) → S’ (Rⁿ)
6. Every elliptic differential operator with constant coefficients is surjective as a map on C^∞(U), for any open set U ⊂ Rⁿ
7. Lidskii’s theorem on trace class operators on L²(Rⁿ)

Grading

The final grade was based on the homework and the project. There were no tests or examinations.

ACTIVITIES	PERCENTAGES
Homework	50%
Project	50%