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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?

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That depends a lot where they are going... take a peek at the curricula of schools that are of interest to you. Snoop around the mathematics (and related) associations (not only in the USA), they probably have suggestions on formation of professionals in the field(s). –  vonbrand May 21 '14 at 23:36
An important question, ... but do you mean to refer to truly gifted students, or to people who "can do the drill" to get through, or barely test out of, the (awful) lower-division courses that are almost universally a "gate"? –  paul garrett May 22 '14 at 0:05
I'm not convinced there is or should be a canonical "ideal sequence". Of course, one should have a firm background in calculus and linear algebra, and sufficient mathematical maturity to take further courses. But other than that I don't see a point in putting an order other than "take the prerequisites to a class before you try to take the class itself". –  Mike Miller May 22 '14 at 1:33
I would think you might like to see the answers at matheducators.stackexchange.com/q/118/128 –  James S. Cook May 22 '14 at 5:20
2-3 mathematics courses per semester is A LOT OF MATHEMATICS COURSES! Even if we use your minimum, 2 courses per semester, that's 16 courses in 4 years. After we add the 2 courses (calculus 1 and calculus 2) that you'd get credit for from having a good BC AP-calculus score, that would be a total of 18 courses, or 54 credit hours of mathematics. Typically (in the U.S.) mathematics majors require 30 to 33 semester hours in mathematics, so you'd be taking MUCH MORE than the typical mathematics major. –  Dave L Renfro May 22 '14 at 14:50

6 Answers 6

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:

  • (1) the notion of what constitutes a proof.
  • (2) the high-level view of mathematics, with an active frontier being pushed daily, sometimes even by undergraduates.
  • (3) This is likely already covered in any carefully considered curriculum, but the historical context in which mathematics emerges adds so much to understanding the content material.
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+1; I have seen (1) and (3) incorporated quite nicely into standard course sequences, but I think the frequent underemphasis on (2) is a shame. –  Benjamin Dickman May 22 '14 at 23:27

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.

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:) $ \textbf{} \textbf{} $ –  Pete L. Clark May 22 '14 at 17:32
You know what I'm talking about :) –  Ryan Reich May 22 '14 at 17:44

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):

Semester 1

  • Mathematical Problem Solving Seminar I (Velleman; Larson; Spivak)
  • Multivariable Calculus (Lang)
  • Either the most advanced physics or computer science course you can take (don't bother with physics if it isn't calculus based)

Semester 2

Semester 3

  • Real Analysis I (Rudin 1-7)
  • Linear Algebra II (Axler)
  • Probability (Ross)

Semester 4

Semester 5

Semester 6

Semester 7

  • Algebraic Topology (Hatcher)
  • Independent Study
  • Independent Research

Semester 8

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.

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I suggest dropping Euclidean geometry. The field is completely dead and knowledge of elementary geometry is basically never useful in contemporary mathematics research. I would wager $50 the majority of math PhDs I know don't have a clue what Ceva's theorem is, for example. –  Potato May 22 '14 at 12:15
I'd nix mathematical logic of the same reason (unless the student in question has a special interest in foundational research). Also, the semester ordering is a bit weird. For example, it's probability not possible to take real analysis I in the spring semester at most schools. It's usually a fall/spring sequence, so it would have to begin on an odd semester. –  Potato May 22 '14 at 12:19
(So in asker's position, I would aim to take it in the first semester of the second year, then the second semester of the undergrad real analysis sequence, then the graduate measure theory-Lebesgue integration-functional analysis sequence the next year.) –  Potato May 22 '14 at 12:21
@Potato I would respectfully disagree with your opinion on mathematical logic. It's not just a foundational course, because, as you have implicitly said, knowledge of it is not useful for doing other mathematics. It has about the same flavor as algebraic geometry (the kind with varieties and equations, not the kind with categories), and indeed, there are interesting connections. Finally, having a broad familiarity with logic has (personally) been enlightening in thinking about math in general. It's a valid field of contemporary research, not a relic. –  Ryan Reich May 22 '14 at 16:27
@RyanReich Sorry, I see that what I wrote implies that it's a relic, and that's of course inaccurate. I should have chosen my words better. I also didn't know about those connections to algebraic geometry, so I admit to outright error there. However, I would still suggest that it is perhaps not as important as getting a solid grounding in analysis, algebra, and topology. –  Potato May 22 '14 at 22:25

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.

Semester 1:

Calculus I (Differential Calculus), Discrete Math, Programming course

Semester 2:

Calculus II (Integral Calculus, Taylor Series), Linear Algebra, Physics I (statics, motion, etc.)

Semester 3:

Multi variable Calculus, Foundations of Math (Logic, proof techniques, set theory), Physics II (E&M, special relativity, maybe basic quantum)

Semester 4:

Differential Equations, Number theory, Probability,

Semester 5:

Theory of Complex Variables, Topology, Statistics

Semester 6:

Chaos/Dynamics, Graph theory, The thing I'm forgetting

Semester 7:

Analysis I (Sequences, more rigorous treatment of calculus), Abstract Algebra (Group theory), Independent research project (first half)

Semester 8:

Analysis II (Sequences of functions, basic measure theory), Abstract Algebra II (Rings), Independent research project (second half)

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If the focus is solely on getting into a good graduate school, then the asker should skip the nonessential classes (physics, chaos theory, dynamics, etc) and instead take graduate classes as soon as possible. –  Potato May 22 '14 at 4:08

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]

Do Carmo's Differential Geometry of Curves and Surfaces OR O'Neill's Elementary Differential Geometry [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]

Lang's Algebra OR Hungerford's Algebra [2-semester Fall-Spring sequence]

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.

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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?

Semester 1:

calculus III, introduction to proofs course, elementary number theory, discrete math

Semester 2:

ODEs, basic matrix theory course, complex analysis, physics I

Semester 3:

PDEs, algebra I, real analysis, linear algebra, physics II

Semester 4:

basic measure theory, algebra II, multivariate analysis, elementary differential geometry

Semester 5:

algebra III, point-set topology, numerical methods, variational calculus

Semester 6:

algebra IV, algebraic topology, operations research, history of math with a focus on nineteenth century math which is still being developed.

Semester 7:

algebra V, manifolds ala John Lee's text, combinatorics, classical mechanics based on variational approach

Semester 8:

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.

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