What is the ideal course sequence for an advanced student of mathematics?

Question

Suppose that you meet a student who:

1. has a firm grasp of algebra and trigonometry and is at least moderately intelligent
2. has read a book such as Love and Math by Edward Frenkel so has some appreciation of math as a discipline including its scope and goals
3. knows they want to be a mathematician

What sequence of mathematical topics would you advise they study. Set aside reality for a moment, this question is not about what is pragmatically best for the student in terms of career. Rather, what course sequence should the unfettered, self-motivated, hard-working, unboundedly-curious student pursue to optimize their understanding of core undergraduate mathematics. Feel free to redefine what the "core" ought to be. Suppose they are just entering university so they have 4 years to study.

Thanks for your thoughts in advance!

Answer 1

Well, there's a fairly standard answer to this question, which is used by most universities in the United States. A student who has completed algebra and trigonometry and wants to be a math major should learn the following subjects:

1. Advanced topics in precalculus mathematics, including logarithms, functions, basic mathematical modeling, advanced algebra techniques, vectors, matrices, conic sections, finite sequences and series, permutations and combinations, some basic probability and statistics, and complex numbers.

2. Single-variable calculus, including limits, differentiation, integration, and infinite sequences and series.

3. At this point, most students would benefit enormously from learning the basics of physics and computer programming.

4. Multivariable and vector calculus, including partial derivatives, the derivative as a matrix, multiple integrals, curvilinear coordinates, vector fields, divergence and curl, Green's theorem, Stokes' Theorem, and the divergence theorem.

5. Linear algebra, including vectors, matrices, row reduction, vector spaces and subspaces, linear independence and span, orthogonality, linear transformations, and eigenvalues and eigenvectors.

6. (Optional) The basics of differential equations, possibly including an introduction to integral transforms, PDE's, and Fourier series.

7. At this point, most students should spend some time working on their formal proofs, and learning to read formal mathematics. Usually, this is best accomplished by learning how to write simple proofs in a relatively concrete mathematical setting, such as number theory, combinatorics, or graph theory.

8. An introduction to abstract algebra, including groups, rings, and fields.

9. An introduction to real analysis, including a rigorous treatment of calculus, as well as topics such as the construction of the real numbers, topology of the real line, metric spaces and norms, pointwise and uniform convergence, and possibly an introduction to measure theory and Lebesgue integration.

10. At this point, most students should continue by learning something about several more topics within mathematics. Some possible choices of topics include:

    • Complex analysis (perhaps mandatory).
    • Point-set topology (perhaps mandatory).
    • Probability and statistics (perhaps mandatory).
    • More linear algebra (perhaps mandatory), including quadratic forms, change of basis, canonical forms, matrix groups, dual spaces, etc.
    • Logic and the foundations of mathematics.
    • Advanced theoretical courses in physics and computer science.
    • Advanced topics in algebra, such as algebraic geometry, commutative algebra, representation theory, or Galois theory.
    • Advanced topics in geometry/topology, including differential geometry, geometric topology, modern geometry, discrete or computational geometry, basic algebraic topology, and dynamical systems.
    • Advanced topics in analysis, such as the theory of PDE's, basic functional analysis, basic harmonic analysis, Lebesgue integration, and so forth.
    • Advanced topics in discrete mathematics.

Answer 2

At least as a counter-point to the other good answers, I must confess that I have misgivings about the standard undergrad math curriculum in the U.S., primarily because I think it presents the subject as a plodding, exaggeratedly fastidious version of the most elementary parts of it... pointedly ignoring genuine motivations, historical motivations, chronology, and too many other aspects, while not allowing itself to mention any substantial contemporary developments.

That is, most often, the "logical order" precept has taken over, and students are taught to worry endlessly over potential pathologies.

At another extreme, there was an older tradition of "engineering math", which went too far in the opposite direction, but did have the virtue of addressing interesting questions, both internal to mathematics and external. Oddly, with the advent of good "math software", it appears that much of what was done by hand by engineers a few decades ago is essentially programmable...

Yes, I am proposing that "rigorous proof" is not the highest goal in mathematics, although, of course, knowing with greater certainty is desirable. Making implicit assumptions overt is good methodology. But being paranoid about technicalities is not good methodology.

A riff I use to explain to the best undergrads I see is that undergrad classes are essentially worthless as preparation for grad school, and the first-year grad courses are scarcely better. That is, quasi-ironically, "we" train beginners into a behavior pattern that we often seemingly endorse for them, but which is not at all what we do.

That is, the possibility of "logical perfection" in mathematics, unlike most other venues, inordinately catches our collective fancies, and become a goal in its own right. (And, possibly, it is worthwhile, but it is not the whole.)

Thus, my caution would be to disbelieve the worldview implicit in undergrad math curricula, and to distrust the choices of focus of it. Better if one can survive in relatively gentle graduate courses, even if they have degenerated (as is often so) into laundry-lists of must-mention topics.

In other words: mathematics is far simpler than as portrayed in undergrad courses.
Best to become acquainted with basics privately and efficiently, unless one truly needs the carrot-and-stick thing.

Answer 3

I'd put a proof-based linear algebra course at the top of the list. I've found that people sometimes overlook the central role of linear algebra. Linear algebra provides not only important tools and results, but the basic language for huge swaths of both pure and applied mathematics, including geometry (Euclidean geometry obviously, but non-Euclidean differential geometry is aptly characterized as "parametrized linear algebra"), abstract algebra, analysis, numerical analysis, etc.

Answer 4

A course that require proofs will be vital, as their high school career probably did not ask them to do enough serious proving. Other than that, there are many directions they can take. Rather than ask for an ideal for all such students, I'd ask for a list of good starting points. It would be great if every such student had a mentor who could help them figure out the ideal sequence for them personally.

Calculus is beautiful and part of every mathematician's vocabulary. There's no sense not taking it if you plan to be a mathematician. But it doesn't have to come first.

A proofs course that gives previews of lots of other courses would be very cool. (I remember reading course descriptions, and being unhappy that I didn't even understand the descriptions of the courses, because those descriptions detailed theorems by name for instance.)

I particularly loved non-Euclidean geometry (in grad school). A smart undergraduate student could take that course and might find it opens their mind.

If I were mentoring such a student, I'd feel them out about whether these courses would be interesting: Number Theory, Non-Euclidean Geometry, Linear Algebra.

Answer 5

In anything remotely computer-related a solid foundation in discrete mathematics (combinatorics, graph theory, ...) is a must.

Number theory shows up in the most unexpected places, more so in computer science and related fields.

Answer 6

So far one important thing missing from the answers here has been some kind of mathematical "play." By play I mean tinkering and struggling with problems to advance mathematical maturity, rather than learning any particular concept or subject area.

I believe at least one of these must be core to an undergraduate math student's plan:

• A research group course, where students split into groups and are tasked with open-ended projects like "The repeating decimal digits of 1/7, 2/7, 3/7, ... are all the same digits but in different orders. What's the deal with that? What other examples of this phenomenon can you find? Why does this happen?" They don't need to discover new mathematics to experience discovery in mathematics.

• A problem seminar in which small groups of students deal with traditional math-contest problems: problems that students are not expected to "already know how" to solve. Math students should not get in the habit of seeing a problem and always expecting to immediately know the first step to take!

• A project course in mathematics: mine was a mathematical modeling in biology course, where a physical situation is given and the students attempt to build a mathematical model, then critique it and talk about the model's strengths and weaknesses.

• An outreach group that runs a math club or math events for younger students. You can do a surprising amount of mathematics with young children -- Euler Characteristic, Nim, Modular Arithmetic, etc are all easaily understood and discoverable by ten-year-olds, and helping with this process of discovery can be excellent experience.

Mathematics problems in the "real world" are often open-ended, and the open-ended problems and questions are the most interesting and inspiring anyway. It was very important for my mathematical development to devote some time each semester to play with mathematics.