Spring 2014, UZH: MAT 636 Ordinary differential equations and fourier series

Time: Thursday 13:00-14:45
Place: Y27H12, Irchel campus

Email: brad.rodgers _put.AT.here_ math.uzh.ch
Course webpage with syllabus: http://www.vorlesungen.uzh.ch/FS14/suche/sm-50689622.modveranst.html
Course discussion site: https://piazza.com/uzh.ch/spring2014/mat636/home

Office hour: By request
Place: Y27K46

Problem session time: Tuesday 14:00-14:45 or Thursday 11:00-11:45
Problem session place: Y27H12, Irchel campus (same as lecture)
Assistant: Nicolas Robles
Email: nicolas.robles _put.AT.here_ math.uzh.ch

Time of final exam: 9:00 - 12:00 (in the morning) on 29.08.2014
Place of exam: Y03G85

Books: For ordinary differential equations we will closely follow the first three chapters of

G. Teschl's Ordinary Differential Equations and Dynamical Systems.
An online copy is available for free on the author's website (halfway down the page).

For Fourier series we will not be closely following any text, but lecture notes will be periodically updated

Lecture notes on Fourier Series.

Some other useful resources (from which we will draw material) include:

E.M. Stein and R. Shakarchi's Fourier Analysis: An Introduction.
S. Krantz's A Panorama of Harmonic Analysis.
T.W. Koerner's Fourier Analysis.
Y. Katznelson's An Introduction to Harmonic Analysis. (slightly more difficult)

Midterm group project:

Midterm - Due ~~May 15~~ May 16, by 5pm

Problem sets:

Homework 1 - Due March 10
Homework 2 - Due March 17
Homework 3 - Due March 24
Homework 4 - Due March 31
Homework 5 - Due April 7
Homework 6 - Due April 17
Homework 7 - Due May 5
Homework 8 - Due May 22
Homework 9 - Not graded. (But please do, in order to review Hilbert space material.)

An approximate calendar of material covered in class

February 20 - Teschl, section 1.1-1.4 (First examples of ODEs and solutions, first order autonomous ODEs, some more complicated explicit solutions)
February 27 - Teschl, section 1.4-1.5 (More examples of explicit solutions, qualitative analysis of ODEs)
March 6 - Teschl, section 2.1-2.2 (Directions fields, very brief discussion of Euler's method, definition of the Picard iteration, Banach fixed-point theorem)
March 13 - Teschl, section 2.2, 2.4 (Proof of the Picard-Lindelöf theorem, Gronwall's inequality and Theorem 2.8)
March 20 - Teschl, section 3.1-3.3 (Examples of autonomous linear first-order systems of ODEs, a review of some linear algebra, matrix exponentials)
March 27 - Teschl, section 3.1-3.3, 3.8 (More matrix exponentials and review of linear algebra, diagonalizable linear systems)
April 3 - Teschl, section 3.1 - 3.3, 3.8 (A review of Jordan normal form, autonomous linear systems in general, qualitative analysis of 2 dimensional systems)
April 10 - Teschl, section 3.3 (Harmonic oscillators: damped and driven), Rudiments of Fourier series
April 17 - Fourier lecture notes, section 1-4 (A review of definitions, the Dirichlet kernel, summability methods, the Fejer kernel)
May 1 - No class, but lecture notes posted later online, section 4-5 (More on the Fejer kernel, and consequences of Fejer's theorem)
May 8 - Fourier lecture notes, section 6, 8, 9 (More general summability kernels, some applications of Fejer's theorem) // Optional: section 7 (du Bois-Reymond's theorem)
May 15 - Fourier lecture notes, parts of sections 11,12 (A summary of convergence results in the past century [without proofs], a review of inner product spaces, an introduction to Hilbert spaces and orthonormal systems)
May 22 - Fourier lecture notes, the rest of sections 11,12 (Complete orthonormal systems, convergence results, Parseval's theorem)
May 29 - No class, but optional lecture notes posted later online, section 13-14 (Riemann's localization theorem, Flat polynomials)

Evaluation:

The final score will be computed with the following weights: 30% homework, 20% midterm, 50% final. The lowest homework grade will be dropped.
What score constitutes a 6, what score a 5, etc. will be set at the end of the semester, however, it will be no tougher than 6 = 95%, and 4 = 70%.
You may work together on homework, and use online and other resources. However, you should say on your homework with whom you have worked, and what external resources you have used, and must write down solutions in your own words. It is a good idea to make a serious attempt at every problem by yourself before consulting a peer or an outside reference.
There will be a midterm and a final. The midterm will be a take-home group project, and the final more traditional.

For the midterm: students will be divided up into groups of three and work on the midterm together, able to use any resources available (including other groups). From the group work, each student will write up one of the solutions, along with a description of the process by which the problem was solved. More information on group projects of this sort, and some examples, can be found here.
The final will be a traditional in-class three hour exam, but it will closely resemble the homework assignments from the course. In particular, at least 2/3 of the problems on the exam will be nearly verbatim homework problems, with only minor variations. The purpose of this is to ease the pressure of a final exam, as well as encourage students to really understand solutions to the homework as opposed to just copying them down from another resource or student.

Piazza:

Piazza is a free resource for discussion related to class material. Go to the website and enroll in MAT 636. You can ask and answer questions here related to homework, material covered in section, or even problems you've come up with on your own. Questions about course policies are better posted on piazza as well, so that answers are public.
Anonymous posting is ok.
Anonymous feedback can be given by posting a private note to the instructors, if you'd like.

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