About my undergraduate mathematics curriculum

Oliver Knill
Eugene Trubowitz lecturing
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 follow-up harmonic analysis course (like Hewitt-Ross), 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
  • 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 (an introduction to 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 [Notes]
  • 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 theorems), Urs Stammbach [Notes]
  • Real analysis (general measure theory on delta rings) Corneliu Constantinescu [Notes]
  • 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 (amazing philosophy lecture series) Paul Feyerabend
5. Semester
  • Functional analysis (Banach/Hilbert spaces, Fredholm theory, operators, spectral theorem) Jürgen Moser
  • Algebra II (rings, fields, representation), Urs Stammbach [Notes]
  • 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) Ernst Specker
  • Banach algebras, (basic harmonic analysis on groups, 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 Preparation notes) Komaravolu Chandrasekkharan
  • Philosophy (Habermas and Adorno), Gerhard Huber
7. Semester
  • Algebraic Topology (cellular complexes, cohomology, [Notes PDF]) Dan Bourghelea
  • Dynamical systems I (stability, symplectic geometry) Jürgen Moser
  • Mathematische software (structure of programming languages, and especially 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 following notes of Paul Rabinowitz, a student of Moser) Jürgen Moser (My preparation notes
  • Russian (beginners course) Felix Ingold
8. Semester
  • Theoretical physics I (quite mathematical approach to Electromagnetism), Walter Hunziker
  • Commutative algebra (ring theory), algebraic geometry) Max-Albert Knus [Notes]
  • Representation theory (of lie groups) Guido Mislin
  • Nonstandard Analysis (building up calculus with Nelson's nonstandard calculus) Hans Läuchli [Notes]
  • 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 (I wrote the lecture notes), Jürgen Moser
  • Dynamical Systems (one dimensional dynamics) Oscar Lanford [Notes]
  • 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 [Notes: Knus Klassische Gruppen]
  • Dynamical Systems II (hyperbolic systems) Oscar Lanford III
  • Logic seminar (complexity theory, I talked about "god numbers" 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 [Notes]
  • Gilbert Baumslag: Topics in combinatorial group theory (the lecture notes appeared as a book 1993)
  • Logic seminar (Specker-Laeuchli): I presented my own nonstandard analysis proof of Fuerstenbergs ergodic theory proof of the van der Waerden theorem.
More graduate courses I took:
  • Jürgen Moser: Celestial mechanics (quite a bit from Siegel-Moser and Zehnder-Moser)
  • Roland Dobrushin: Topics in statistical mechanics (models like the Ising model)
  • Eduard Zehnder: Selected topics Symplectic Geometry (mostly symplectic capacities, following the emerging Hofer-Zehnder book)
  • Krzysztof Gawedzki: Quantum field theory (Conformal field theory, QFT,Virasora,Wess-Zumino-Witten,Moduli etc) [Notes]
  • Joel Feldman: Mathematical Methods in Many Body Theory (Fermi sea, Feynman diagrams, Renormalization, Superconductivity) [Notes]

My first desk in the analysis assistant group was in a larger office with several assistants, like Marcus Heusler, Kurt Pagani, a computer corner, where the logician and CS guru Beat Scherer would demonstrate the latest. It was there that I played first games like Tetris, see the Macintosh speak or see pre-versions of Mathematica. I at the time still had my Atari ST. Sometimes, the office was joined by postdocs like Wilhelm Klingenberg Jr.. To the right my second grad school desk at ETH center in the analysis assist group. I was most of the time with Martin Flucher, Lutz Wilhelmy, and Moritz Adelmeyer. There was also Thomas Wurms or Bernhard Schmithüsen .
Click on one of the pictures to the right to see it larger.
Here is a better photo in my second office:
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. 2021: added two pictures from first assistant group office.