What is the perfectly (maybe unrealistically) ideal undergraduate sequence for a undergraduate majoring in pure mathematics who takes 2-3 mathematics courses per semester assuming a strong AP Calculus BC background and wishes to go on to graduate school for a PhD in mathematics?
I wonder if we (or some formal international body) should rethink the entire curriculum? Aeryk's very logical and reasonable curriculum is focused totally on content, as opposed to what we want the students to learn along the way. In other words, it is not only sequence that matters!
I am particularly concerned about three aspects that disappear in content lists:
Taking inspiration from @Aeryk, here is my (as realistic as possible) dream proposal for a good student at a good university but without access to graduate level classes (such as at a LAC):
In the last two or three semesters, one may wish to take a fourth math class that piques ones interest.
These are just my initial thoughts -- please offer any criticism and suggestions in the comments.
Oh, this is quite straightforward. First, go to a school with quarters rather than semesters: this gives you 50% more classes to take. Then, take every upper-division class and anything marked as "honors", in order of prerequisites and interest. Take graduate classes if you run out.
For whatever it's worth, below is roughly the schedule followed by the 5 or 6 top math students over a 4-year period where I was an undergraduate in the late 1970s. Two went to Berkeley, one went to Princeton, one (who was a top 30 Putnam scorer) left math for riches in computer science, and one went to MIT for a Ph.D. in mechanical engineering. Incidentally, I know of at least one person from this time period who DID NOT follow this path (took advanced calculus in 3rd year, maybe two graduate level classes in 4th year) who is currently a full professor at a Group II university (I just checked).
I'm only listing math courses below, but most took several computer science and physics courses. Two of them, maybe three, took pretty much all of the undergraduate theoretical physics courses (2-semester sequence in classical mechanics, 2-semester sequence in electrodynamics, Saxon level quantum mechanics, solid state physics, optics) along with two or three graduate physics courses.
In some cases the text varied in a predictable way depending on who taught the course. For example, Professor X would use Do Carmo in years n, n + 2, n + 4, etc. and Professor Y would use O'Neill in years n+1, n+3, etc.
1st Year (Fall, Spring) Apostol's Calculus, Volume 2
2nd year (Fall)
Hoffman/Kunze's Linear Algebra
Taylor/Mann's Advanced Calculus (first few chapters and the last part on sequences, series, improper integrals, etc.)
2nd Year (Spring)
Standard cookbook ODE course [some may have skipped this]
Fleming's Functions of Several Variables [Special pull-out reading course for top 2 or 3 students each year taking the second semester of the Taylor/Mann Advanced Calculus sequence.]
3rd and 4th Years (order taken varied; only one or two took all of these)
Royden's Real Analysis [offered in Fall]
Munkres' Topology (entire book covered except for Stone-Cech compactification, some of the metrization results, and dimension theory) AND much of Massey's Algebraic Topology: An Introduction [offered in Spring]
Hirsch/Smale's Differential Equations, Dynamical Systems, and Linear Algebra [offered in Fall]
Warner's Foundations of Differentiable Manifolds and Lie Groups OR Spivak's A Comprehensive Introduction to Differential Geometry (Volume 1) [offered in Spring]
Besides courses from those above, these students took one or two other "special topics" graduate courses that were offered at irregular times, such as calculus of variations and ergodic theory.
Here's my opinion/wish. For some of the classes the chronological order doesn't matter. For some it probably does and/or I might not have the best order.
Calculus I (Differential Calculus), Discrete Math, Programming course
Calculus II (Integral Calculus, Taylor Series), Linear Algebra, Physics I (statics, motion, etc.)
Multi variable Calculus, Foundations of Math (Logic, proof techniques, set theory), Physics II (E&M, special relativity, maybe basic quantum)
Differential Equations, Number theory, Probability,
Theory of Complex Variables, Topology, Statistics
Chaos/Dynamics, Graph theory, The thing I'm forgetting
Analysis I (Sequences, more rigorous treatment of calculus), Abstract Algebra (Group theory), Independent research project (first half)
Analysis II (Sequences of functions, basic measure theory), Abstract Algebra II (Rings), Independent research project (second half)
Ideally, well, in retrospect, call multiple disciplinary studies bluff and use their flexibility to eliminate the fluff. All the sudden, we have time for 4-5 technical courses a semester. A hard path, but, if we stay on it where does it take us?
calculus III, introduction to proofs course, elementary number theory, discrete math
ODEs, basic matrix theory course, complex analysis, physics I
PDEs, algebra I, real analysis, linear algebra, physics II
basic measure theory, algebra II, multivariate analysis, elementary differential geometry
algebra III, point-set topology, numerical methods, variational calculus
algebra IV, algebraic topology, operations research, history of math with a focus on nineteenth century math which is still being developed.
algebra V, manifolds ala John Lee's text, combinatorics, classical mechanics based on variational approach
geometry of differential forms as in Shigeyuki Morita's text, analytic number theory, chaos and dynamics, electromagnetism with vector calculus approach
Now, I had one semester which was like the ones I list above. Once upon a time, I took a semester with 5 pure courses in math and physics. It was a pure joy. However, reality intervened and after that, the need to satiate graduation requirements diluted the joy with courses from other schools. The sequence I list above is slanted towards a particular direction of differential geometry or perhaps mathematical physics. If you were otherwise inclined, you would want to take more number theory or programming much earlier. By algebra I,II, III, IV, V I intend a elementary treatment from something like Gallian followed by a few semesters digging through Dummit and Foote. Also, if you could find a few foundations courses in set theory or categories it would be nice at some point.
Honestly, it really depends on what you want to do. Also, it depends strongly on what can be done at your school if you go the traditional route. The path of independent study is worth investigating if your school leaves you wanting more.
Like it or not, summer research is a good move for the sake of seeing outside your school. Moreover, earlier is better, some programs are looking for students without too much course work. Those could be a good gateway. It is not easy to get into these so plan to do some research about where and what you want to attempt. Personally, I ignored an opportunity I had to go with a professor to Los Alamos and work in mathematical physics. Who knows, if I had been less of an idiot and taken the opportunity I might still be in physics. But, life is good, just saying, don't pass up such opportunities lightly.