Seminar in Analysis 18.104

Spring 2013

Instructor: Kiril Datchev, room 2-173, Office hours by appointment.

Class meetings: Tuesdays and Thursdays 2:30-4:00 in room 2-131.

Textbook: F.G. Friedlander and M. Joshi, An Introduction to the Theory of Distributions, second edition, 1999. Beware of errata.

Recommended reading: R.S. Strichartz, A guide to distribution theory and Fourier transforms. J.J. Duistermaat and J.A.C. Kolk, Distributions: Theory and Applications. This book is available electronically from the MIT library here. For inspirational reading, consult The Study of Mathematics by Bertrand Russell.

Topics covered: Distribution Theory. Test functions, distributions, differentiation, support, convolution, Fourier transform. Applications to differential equations, mathematical physics. Other topics according to student interest, such as Sobolev spaces, Fourier-Laplace transforms, Schwartz representation and kernel theorems, singular supports, elliptic regularity, wavefront sets, distributions on manifolds, Fourier series, lattice point asymptotics, pseudodifferential operators.

Grades are determined as follows: 50% Lectures, 25% Final Paper, 25% Homework.

Attendance is mandatory. Your grade will be dropped by one letter for every three unexcused absences (not counting absences in the first week).

Lectures. Each student will give three or four lectures of 40 minutes each during the semester. Lectures will be evaluated based on clarity, organization, preparedness, and improvement over the course of the semester. Students will give a practice lecture to me before their first lecture.

Homework is due by the beginning of class and must be written in Latex (either handed in or emailed to me as a pdf). You may use the tex file of the assignment as a template or use your own template. Feel free to ignore the hints provided if you see a better way to do a problem. If you want to include figures (recommended!) and have trouble generating them electronically, you may draw them in neatly by hand on your printed assignment. Collaboration is encouraged, but you must write up your homework on your own. Late homework may be handed in by the following class for half credit.

The final paper is an exposition of a topic related to distribution theory beyond the material covered in the course. Here are more instructions and a list of suggested topics.

Remarks: The textbook requires more mathematical sophistication than most introductory texts in analysis. We will take advantage of the small size of the class to make sure everyone manages the jump successfully, but be ready to make the necessary effort! Some comments made in passing are above the level of the main thread of the text, and while these can be interesting starting points for further directions of study (for example in the final project), it is important not to let yourself get stuck. Email me to set up a meeting if you're having trouble.

Date Speaker Material (*indicates tentative) Homework due
2/5 Kiril §1.2 none
2/7 Kiril §1.3 none
2/12 Yair §1.4 through p11 tex pdf
Saul §1.4 from p12
2/14 Amol §1.5 tex pdf
Sam §2.1
2/21 Kiril §2.2,3,5 tex pdf
2/26 Zipei §2.4 tex pdf
Suyan §2.7
2/28 Sruthi §3.1 tex pdf
Sung Gi §3.2
3/5 Kiril §4.1,2 tex pdf
3/7 Soohyun §4.3 tex pdf
Aryan §4.3
3/12 Gabriel §5.1 tex pdf
Kiril §5.2
3/14 Kiril §5.3 tex pdf
3/19 Yair §5.4 look at the paper topics
Suyan §5.4
3/21 Sruthi §8.1 look at the paper topics
Amol §8.2
4/2 Kiril §8.3 work on paper
4/4 Sung Gi §8.4 tex pdf
Zipei §8.5
4/9 Soohyun §8.5 tex pdf
Gabriel §8.6
4/11 Kiril §8.6, §9.1 tex pdf
4/16 PATRIOT'S DAY -- NO CLASS First draft of paper due
4/18 Sam
4/23 Saul
4/25 Kiril
4/30 Saul
5/2 Amol
5/7 Aryan
5/9 Sam
5/14 Yair
Sung Gi
5/16 Gabriel