My undergraduate courses at ETH

About my undergraduate mathematics curriculum

Oliver Knill

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).

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 some more abstract algebraic geometry), a solid harmonic analysis course, a course on stochastic differential equations, and advanced combinatorics and graph theory course.
Exams In the first two years, all exams were both oral and written. Later on, most exams were oral only.

1. Semester Analysis I (single variable calculus) Hans Läuchli Linear Algebra I (starting with linear equations) Ernst Specker Geometry (elementary Euclidean geometry, symmetries, groups), Max Jeger Numerics (with Pascal programming language) Peter Läuchli Astronomy (a basic course learning terminology, structure of the universe etc) Harry Nussbaumer	2. Semester Analysis II (multi-variable calculus), Hans Läuchli Linear algebra II (starting with diagonalization, and applications) Ernst Specker Physics I (mechanics, electromagnetism) Hans Leisi Numerics and programming II (numerical algorithms), Peter Henrici
3. Semester Analysis III (complex analysis and Fourier theory) Eugene Trubowitz Topology (axiomatic point set topology) Erwin Engeler Logic and set theory (foundations of logics and set theory) Ernst Specker Number theory (elementary number theory, geometry of numbers) Komaravolu Chandrasekkharan Physik II (heat, optics) Hans Leisi Theoretical Mechanics (Lagrangian and Hamiltonian dynamics, integrable systems until Arnold theorem) Jürg Fröhlich	4. Semester Analysis IV (differential equations, basic Lie groups, applications) Eugene Trubowitz Probability and statistics (i.e. limit theorems, mathematical statistics) Hans Föllmer Algebra I (theory of groups up to Sylow), Urs Stammbach Real analysis (measure theory on delta rings) Corneliu Constantinescu Geometry II (transformation groups, finite and projective geometry) Max Jeger Physics laboratory, (structured afternoon labs, with many different real experiments) Hans Rudolf Ott Limits of knowledge (philosophy lecture series) Paul Feyerabend
5. Semester Functional analysis (Banach/Hilbert spaces, Fredholm theory, operators, spectral theorem) Jürgen Moser Algebra II (rings and fields), Urs Stammbach Differential geometry (curves and surfaces until Gauss-Bonnet) Max Jeger Proseminar Riemann surfaces (I presented a monodromy theorem for Riemann surfaces) Eugene Trubowitz Philosophy (reading and discussion) Gerhard Huber	6. Semester Functional analysis II (partial differential equations), Jürgen Moser Lie groups and algebras (all classical lie groups and algebras), Guido Mislin Model theory, (forcing) Hans Läuchli Banach algebras, (harmonic analysis, von Neuman algebras, C^algebras) Christian Blatter Theoretical computer science (complexity, languages, models*), Erwin Engeler Proseminar number theory (I presented a gap theorem of Polya on Dirichlet series) Komaravolu Chandrasekkharan Philosophy (Habermas and Adorno), Gerhard Huber
7. Semester Algebraic Topology (cellular complexes, cohomology) Dan Burghelea Dynamical systems I (stability, symplectic geometry) Jürgen Moser Mathematische software (building up languages Lisp) Erwin Engeler Mathematical logic (formal theory, predicate calculus, models) Ernst Specker Axiomatics (Goedel) Hans Häuchli Quantum mechanics (for physisists) Klaus Hepp Proseminar dynamical systems (I presented stability theorem for multidimensional complex dynamical systems) Jürgen Moser Russian (beginners course) Felix Ingold	8. Semester Theoretical physics I (electromagnetism), Walter Hunziker Commutative algebra (ring theory, algebraic geometry) Max-Albert Knus Representation theory (of lie groups) Guido Mislin Nonstandardanalysis (building up calculus with nonstandard calculus) Hans Läuchli Dynamical systems II (differential equations, special topics), Jürgen Moser Proseminar dynamical systems (I presented the Poincare-Birkhoff theorem) Jürgen Moser Proseminar Lie groups (I presented a Theorem of Engel) Guido Mislin
9. Semester Topics in the calculus of variations, Jürgen Moser Dynamical Systems (one dimensional dynamics) Oscar Lanford Theoretical computer science (complexity theory) Erwin Engeler Algorithms Ernst Specker Russian, Felix Ingold Logics seminar (I presented a paper on the deciding problem of graph isomorphisms), Ernst Specker	10. Semester Classical groups (advanced algebra) Max-Albert Knus Dynamical Systems II (hyperbolic systems) Oscar Lanford III Logic seminar (complexity theory, I talked about the diameter of finitely presented groups), Ernst Specker Diploma theses, (The Störmer problem) Jürgen Moser
Some graduate courses, I took: Michael Struwe: Nonlinear wave equations Anthony Tromba: Teichmüller Theory in Riemannian Geometry David Ruelle: Dynamical Zeta functions Gilbert Baumslag: Topics in combinatorial group theory	More graduate courses I took: Jürgen Moser: Celestial mechanics Roland Dobrushin: Topics in statistical mechanics Eduard Zehnder: Symplectic Geometry Krzysztof Gawedzki: Quantum field theory

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.