Where to go after calculus?

Solution 1:

A good History of Math course would probably be enjoyable, and give you a good idea of what things are and where they lie.

From the mathematical point of view, Linear algebra is often a good "first advanced mathematics course", a good jumping-off point: it is still concrete enough that you won't get lost in a sea of abstraction (a possible issue with abstract algebra depending on how it is taught), cover entirely new ideas ("advanced calculus" can feel like you are just re-treading the same ground you already know, and depending on the specific topics analysis might also feel like it), but it should make you work through proofs and concepts in a way with which you probably have not done so far. In addition, linear methods will show up all over the place later on, so it would prove useful.

In that same vein, Number Theory can be a really good "first abstract course in mathematics", while sticking close to things you are very familiar with (the integers and rationals) while also probably delivering some exciting surprises. It often surprises a lot of people just how much of mathematics arises out of number theory (complex analysis and abstract algebra, to name just two).

If you want to stick to the applied side, differential equations is a good place to go as well. Linear algebra would be useful there, though.

So, I would suggest linear algebra or number theory first (if you can also get a good history of math course, do that as well), then decide if you want to go towards abstraction (in which case, head to abstract algebra, mathematical analysis, or whichever of linear algebra or number theory you did not take) or more towards applications (differential equations, a good advanced probability/statistics course, or a discrete mathematics course).

It is indeed the case that mathematics starts branching out, but some of the most interesting things happen where the branches meet; it would be ideal to be able to take a good one semester or one year sequence in the major areas (analysis, algebra, differential equations, topology, logic/set theory, number theory), then go on to more advanced courses in whichever area(s) you find interesting. But the truth is that this is very hard to do: not only would such a wide choice not be available except in the largest universities, but it would also mean a lot of your time. I did my undergraduate in Mexico, where all I did in college was mathematics courses, and it basically took six semesters before that had been covered (in addition to the calculus sequence, a linear algebra sequence, an abstract algebra sequence, a mathematical analysis sequence, differential equations, discrete mathematics, complex analysis, probability and statistics, plus some other stuff to "fill in the corners"; it would be barely possible to do it in two years if you are not taking the calculus sequence, but not if you are also taking other coursework as you would be in most institutions in the United States).

Solution 2:

Large parts of mathematics as it has evolved in the 20th century are quite independent of calculus. Game theory, graph theory, combinatorics, discrete mathematics, etc. are often courses which are taught without a specific Calculus prerequisite. The order in which one studies mathematics is to some extent related to what one's goals and "natural talents" are. For example, there are parts of geometry which don't require calculus (directly) while there are other parts of geometry where using Calculus methods are very central. If one majors in mathematics typically there is a "recommended route" of courses at the school one is studying at.

Solution 3:

If you're interested in building on your calculus knowledge in a way that builds links to many other areas of mathematics, a natural direction to go would be to take some linear algebra and a differential geometry course. From there you could take some analysis and then a manifolds course. At that point the subject of Lie groups acts as something of a unifying picture for all of the above said courses -- it fits everything together in a very pleasant way and gives you many pleasant connections between group theory, linear algebra and analysis. Stillwell's book "Naive Lie Theory" is something you could pick up right now and start reading. When reading the book you'll likely be able to spot what you're comfortable with and what you aren't, and go from there.

Solution 4:

One of the disappointments experienced by most mathematics students is that they never get a course in mathematics. They get courses in calculus, algebra, topology, and so on, but the divison of labor in teaching seems to prevent these different topics from being combined into a whole. In fact, some of the most important and natural questions are stifled because they wall on the wrong side of topic boundary lines. ... Thus, if students are to feel they really know mathematics by the time they graduate, there is a need to unify the subject.

from Stillwell - Mathematics and Its History

So you could take this time to study some of the history of mathematics. Another thing which is useful is to study some formal logic (the stuff that makes proofs) - since (at least in my experience) this is not taught very much in classes.

Solution 5:

I do not know what "Calculus 1-3" entails, but if the terms "complete ordered field" and "least upper bound property" don't mean anything to you, I'd start with a first course in real analysis. The usual stuff: Cauchy sequences, absolutely and conditionally convergent series, the $\varepsilon$-$\delta$ definition of continuity, the intermediate value theorem, a rigorous construction of the Riemann integral using upper and lower sums$\ldots$

All (at least non-nonstandard) analysis flows from this, up to and including metric spaces, topology (to a lesser degree admittedly), differential geometry, complex analysis, functional analysis and meassure theory.

Paraphrasing the late Randy Pausch: Fundamentals. Fundamentals. Fundamentals. You need to have the basics straight or the fancy stuff ain't gonna work.

From there I'd look at linear algebra. As has been mentioned earlier, it's nice and concrete but still very useful and shows up in a lot of places. For example in abstract algebra when studying field extensions(a).

Speaking of abstract algebra, it takes up an interesting place in undergraduate mathematics. I find it hard to explain, but it's like it's sitting in a corner off by itself doing its own thing - at least until you get to for example algebraic topology. But algebraic topology isn't usually taught to undergraduates (at least that I know of), so algebra feels rather disconnected from the rest of an undergraduate mathematics education.

So what it boils down to is that I'd stay away from abstract algebra beyond a first course (because all undergraduate math students need to at least know groups, rings and fields) unless you either 1) loved the first course, 2) find out that algebraic tools are needed for something else you like, or 3) the rules say you must do more algebra.

(a) Though probably not in a first course in abstract algebra since Galois theory can't really be taught until after you've done the basics of groups, rings and fields which take up at least a semester.