Calculus III Lecture of Eugene Trubowitz on differential equations. Photo by Oliver Knill with a Minox EC camera. The Trubowitz lectures were especially fun and entertaining. He covered unusual material (an example). Here are notes of my colleque Urs Barmetteler of a linear algebra course and notes taken by Pius Zängerle of a Mathematical physics course by Trubowitz Look at the breath of these introduction courses! Added May, 2013: some outlines for the intro lectures. 
Curriculum How does an optimized mathematics curriculum look like? While this is certainly a matter of the nature of the college, the student distribution and also taste, I describe briefly the program I followed as an undergraduate at the Eidgenössische Hochschule Zürich" (ETH). In retrospective, the program (before the Bologna process) looks close to optimal to me now. I especially appreciate now the exposure to a wide range of mathematics as an undergraduate. There was no major field of mathematics which was left out. Below is a list reconstructed from transcripts and memory. 
Terminology
Proseminars were seminars, where the students would present material from
books or papers. They were mostly guided by senior faculty, (who are assisted by their
graduate students and FIM guests) and were quite intense. They ressemble
Tutorials at Harvard but were
almost exclusively run by senior faculty assited by graduate students. I suspect now,
that Proseminars were also used for scouting course assistants or graduate students because
watching somebody present difficult material on the blackboard can give good pointers about
teaching skills (preparation, presentation and interaction with the audience). Different flavors of studying mathematics: there was almost no choice in the first two years. Everybody practically had the same courses. After that, the student could follow into one of the three tracks "theoretical physics", "computer science", or "probability and statistics". This choice would not narrow down the list of courses one could take, but it defined a few requirements of proseminars and courses in that field. I myself choose the "theoretical physics" track. 
Semesters
The usual ETH undergraduate curriculum is 8 semester. Many of my friends did 9 or
10, often due to similar reasons than mine: since I did 40 weeks of military service
during that time, taught for several weeks at a high school in Winterthur,
and was a course assistant in calculus and computer science laboratory as an
undergraduate, I needed 10 semester to get all the requirements (*).
This included the final diploma theses, which had to be written during a 4
month period. (*) Programming insane hours on the early macintosh computers, spending months in the mountains trekking a summer through Europe (often by foot), biking through the mountains in Switzerland, or spending many afternoons on our boat on the rhein did not help neither). 
Choice
As a student, I found it relaxing to start with a well structured program and not have to
build my own calendar in the first two years. Such a well structured program was
possible at ETH because students know their concentration when enrolling. Also, for
the first four semesters, all physics and mathematics students took courses together.
I also liked not to have to choose the level of difficulty of the courses
nor to fret what kind of mathematics (or  a much more stressful question what level of mathematics)
could turn out to be useful later: the decision whether it is important to take a course on
set topology in the 3. semester or whether to bother with logic at all is not obvious at
that time. Even so we had learned calculus already in high school, every ETH student had to
go through calculus course from scratch. An advantage of such a structured program was
that courses fitted together perfectly. There was no unnecessary overlap, no need
to repeat or the need of advising. Because physics and mathematics students have the same
courses in the first two years, also the physics courses worked well together.
The instructor in each course knew precisely what the background of the class was.
In physics II for example, the entire class had been exposed to integral theorems and linear
algebra so that one could deal with Maxwell equations or wave equations right away.
The last two years of my undergraduate studies were less structured. I had to deal
more with time management issues as well as
the paradox of choice. Essentials In the following list, the courses, I consider essential are marked reddish. There are still gaps: I would have appreciated an additional, less algebraic but more geometric algebraic geometry course (also in military, I only learned mostly abstract algebraic geometry a la Hartshorne), a followup harmonic analysis course (like HewittRoss), a course on stochastic differential equations, and a graph theory course (even so some spectral graph theory and graph combinatorics had appeared in linear algebra and graph complexity questions in a seminar). Exams In the first two years, all exams were both oral and written. Later on, most exams were oral only. 
1. Semester

2. Semester

3. Semester

4. Semester

5. Semester

6. Semester

7. Semester

8. Semester

9. Semester

10. Semester

Some graduate courses, I took:

More graduate courses I took:

Back to Olivers Homepage, put online July 2, 2006, 2009: Marked every course in red, which I think every mathematician should take for at least one semester. 