Ask HN: How to self-study mathematics from the undergrad through graduate level?

There are plenty of people who devote several years of their life to studying, and must pay not only their living expenses but tuition fees as well. In my opinion, those are the people who you should be asking “how did you manage this”.

I have some opinions on the question in the OP: I'm heavily self-taught from independent study in math; that study helped me with, and at times was part of my good career in, applied math and computing before my Ph.D. in pure/applied math and helped a lot for my Ph.D.

For a curriculum:

(1) Do the standard high school math, Algebra I, Plane Geometry (based on proofs), Algebra II, Trigonometry, and Solid Geometry (based on proofs). In each of these, it is sufficient (A) to work most of the more challenging exercises and (B) try to think a little about what is going on. For (C), it would be good occasionally, say, 1-4 times a year, to chat about your progress with a good mathematician, Ph.D., likely a college prof. To find a Ph.D. who will give you an hour or so per visit, maybe do some networking. If you don't like the first choice, e.g., if they are not encouraging and helpful, then try a different Ph.D. mathematician.

(2) Do Analytic Geometry and Calculus. Use a good college textbook. Since highly polished texts have been widely available for decades, there's no need to pay big bucks for new copies of the latest. So get a good used text. Better yet, get 1-4 good texts, use your favorite one as your primary source and the others for more content, alternate descriptions, evidence of what is more/less important, etc.

Sets: Now in an important sense, essentially all of math is heavily about sets. For first calculus, you don't need to know much about sets; the following should be nearly sufficient:

By 1900 or so, a lot of math was known, but there was a nagging, somewhat philosophical, question of what math really is. The answer of pure math was to construct a foundation in, say, the deep basement, that would answer the question but not much change the rest. The answer was to start with sets and then define everything else in terms of sets. In particular do some somewhat tricky and obscure work to define some sets that look, work, walk, talk, etc. like the numbers on the line, the real numbers. We don't think of the real numbers this way, but the effort addressed the philosophical question -- or, if I can get you to believe in sets, then I've got you for the rest of math. Much of the early work on sets was from G. Cantor. Yes, soon there were some issues from Bertrand Russell and then Kurt Gödel and, later, Paul Cohen.

For a long time in math, you can regard a set as a conceptual (imaginary) collection of some kind, e.g., all the peas in a little jar in the back of the refrigerator, all the people with an iPhone, all the real numbers on the line, all the water molecules in the earth's oceans, all the parabolas, all the trials you might do in a lab experiment, and much more -- really, from the philosophical approach to what math is for all of math. Or intuitively sets are the containers for what we are thinking about.

Functions: Much of math is about functions. E.g.,

     f(x) = 2x^2 + 3x - 4

     "Think of something; call it x."

     "I suspect there is a number such
     that these conditions hold; I want to
     find if there is such a number and if
     so what its actual numerical value
     is."

E.g., for this set stuff, a function gets defined in terms of sets, e.g., the set of all ordered pairs

     (x, f(x))

So, with the ordered pairs

     (x, f(x))

Of course, in actual work, especially in applied math, nearly no one actually thinks of a function as a set of ordered pairs.

Completeness: There is an old joke, partly appropriate, that

     "Calculus is the elementary
     consequences of the completeness
     property of the real number system.".

     (p/q)(p/q) = 2

     (p)(p) = 2(q)(q)

Well, a big deal about the real numbers is that they are complete; i.e., intuitively, if something converges, then there is something to converge to. Completeness is a big deal because it generalizes, especially to Hilbert space that you may come to.

So, in calculus often we make approximations that can become as accurate as we please, and we want the completeness property to know that what these approximations approach, limits, can exist.

E.g., with the real numbers, if we make more and more accurate approximations to the square root of 2, then we know that our approximations will converge to the actual square root of 2 which DOES exist as a real number.

Sets get to be quite important in math in the last years of college math and beyond. But for calculus, nearly all of that subject, especially what learn in a first calculus course, was quite solid well before G. Cantor. So, in practice, when studying first calculus, we don't see much about sets. So, for first calculus, sets remain mostly a topic in the deep basement that addresses an old philosophical issue. So, for first calculus, sets are no big issue.

Analytic Geometry: Here you study the conic sections. They are a bit amazing and important in physics, mechanical engineering, and more and something definite to work with while learning calculus. So, imagine a cone, say, an ice cream cone. Take two of them that are the same and put the points together so that they are both on the same axis through the points (I'll let you get a precise definition of axis). Now imagine a sharp sword, say, as in the old John Belushi Samurai Tailor skit and use the sword to slice the cones and look at the cut edges. Right, we assume that the sword moves in a plane. Depending on where you slice, you will get (A) just a point (where the two cones touch together), (B) two lines crossed as in an X (the cross in the X is at the point where the two cones come together), (C) a circle (we are awash in circles), (D) a parabola (to a fairly good approximation, a grand slam baseball follows a parabola), (E) an ellipse (to a quite good approximation, each planet goes around the sun in an ellipse), (E) the two parts of a hyperbola (an electron shot at a negative charge will follow one of the halves of a hyperbola as it avoids the negative charge).

First Calculus in a Nutshell: Then for first calculus, there are two big topics, differentiation and integration. Differentiation is finding rate of change, intuitively the slope (as in high school algebra) of a tangent line to the graph of a curve (defined in terms of a function with domain some or all of the real numbers). E.g., for time t, let d(t) be the function that gives us our distance from home at time t. Then the derivative of function d at time t is the velocity, say, function v, at time t. So, in a car, the odometer gives d and the speedometer gives v. That is, at time t we have traveled distance d(t) and are moving a velocity v(t). Here to be simple, I do not make a careful distinction between what physics calls speed versus velocity.

What differentiation does, integration reverses, undoes. So, we can use integration on function v to recover function d. We have now introduced the fundamental theorem of calculus.

Right, v is the function, and v(t) is its value at time t, but in first calculus it is common to drop the distinction between v and v(t) and just say "the function v(t)" and mostly avoid mentioning v by itself.

Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and where each frequency w is a whole number multiple of a fundamental frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes.

The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers.

I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of

Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957.

Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly enough for what physics and engineering do with vector analysis and Stokes theorem.

In H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis, can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations.

As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes.

In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory.

From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant".

For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some.

One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as significant based on what it does for the real problem and not just its contribution to pure math.

Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more.

In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year.

For a little on how to do the original parts of math research, there is the A. Wiles comment:

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...."

That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you.

Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.