18.155 Differential Analysis -- Fall 2007

Schedule: Tue/Thu 11-12:30, room 2-135
Office hours: Mon 2-3, Wed 3-4 or by appointment, office 2-277
Contact: Katrin Wehrheim ( katrin (guess what) math.mit.edu )

Homework:

Problem Set 11, due Thursday 12/6 in class.

Problem Set 10, due Thursday 11/29 in class.

Problem Set 9, due Tuesday 11/20 in class or Thursday 11/22, by email.

Problem Set 8, due Thursday 11/15, in class.

Problem Set 7, due Thursday 11/1, in class.

Problem Set 6, due Thursday 10/25, in class. Some Solutions

Problem Set 5, due Thursday 10/18, in class.

Problem Set 4, due Thursday 10/11, in class. Solution to Problem 3 by Anand Deopurkar

Problem Set 3, due Thursday 10/4, in class. Some Solutions

Problem Set 2, due Thursday 9/27, in class.

Problem Set 1, due Thursday 9/20, in class. Solutions

zeroth "assignment" due Friday 9/7, midnight: Email me

A brief sketch of the definition of the Riemann integral, as you learned it.

Whether or not you have previously studied the Lebesgue integral. If so, email a rough sketch of the definition.

This just needs to be a few lines. I'm looking for key words like step function, partition, approximation, zero set, measure, sigma-algebra, simple function, measurable function.


Topics:
Lebesgue integral, L^p spaces
Distributions, Fourier transform and inversion
Sobolev spaces, embedding theorems, Rellich's theorem
partial differential equations with constant coefficients,
elliptic, hyperbolic, parabolic operators, and their fundamental solutions
Banach spaces, Fredholm operators
Hilbert spaces, spectral theorems, discrete and continuous spectrum

Literature:
Lecture Notes by Richard Melrose
Script on construction of Riemann and Lebesgue integral
Script on Lp-multipliers and Dirac sequences
W. Rudin, Real and complex analysis, McGraw-Hill, 1987.
R.A.Adams, J.J.F.Fournier, Sobolev Spaces, Elsevier 2003.
S.Lang, Real and Functional Analysis, Springer 1993.
E.Kreyszig, Introductory Functional Analysis with Applications, Krieger 1989.

Grades:
The final grade will be based on the homework. There will be no tests or examinations.