It is important when studying mathematics to do so with the following perspective
Mathematicians Allow Useless Non-computable Fantasy Objects
Mathematicians often choose to live in a world where the axiom of choice is true for sets of size the continuum. This is idiotic for many reasons, even for them, but it is especially idiotic for physics. There are easy intuitive arguments that establish that every set has a volume, or Lebesgue measure, and they go like this:
Given any set S in a big box B, choose points randomly and consider when they land in S. In the limit of many throws, define the measure of S to be the volume of B times the fraction of points which land in S. When this works, and it always works, every set is measurable.
This definition is not allowed in mathematics, because the concept of randomly choosing a point requires taking a limit of the random process of choosing the digits at random. The limiting random process must be defined separately from the approximation processes within usual mathematics, even when the approximations almost always converge to a unique answer! The only reason for this is that there are axiom of choice constructions of non-measurable sets, so that the argument above cannot be allowed to go through. This leads to many cumbersome conventions which inhibit understanding.
If you read mathematics, keep in the back of your head that every set of real numbers is really measurable, that every ordinal is really countable (even the ones that pretend to be uncountable collapse to countable ones in actual models of set theory), and that all the fantasy results of mathematics come from mapping the real numbers to an ordinal. When you map the real numbers to an ordinal, you are pretending that some set theory model, which is secretly countable by the Skolem theorem, contains all the real numbers. This causes the set of real numbers to be secretly countable. This doesn't lead to a paradox if you don't allow yourself to choose real numbers at random, because all the real numbers you can make symbols for are countable, because there are only countable many symbols. But, if you reveal this countability by admitting a symbol which represents a one-to-one map between some ordinal and the real numbers, you get Vitali theorems about non-measurable sets. These theorems can never impact physics, because these "theorems" are false in every real intepretation, even within mathematics.
Because of this, you can basically ignore the following:
- Advanced point set topology--- the nontrivial results of point-set topology are useless, because they are often analyzing the choice structure of the continuum. The trivial results are just restating elementary continuity properties in set theoretic language. The whole field is bankrupt. The only useful thing in it is the study of topologies on discrete sets.
- Elementary measure theory: while advanced measure theory (probability) is very important, the elementary treatments of measure theory are basically concerning themselves with the fantasy that there are non-measurable sets. You should never prove a set is measurable, because all sets are measurable. Ignore this part of the book, and skip directly to the advanced parts.
Discrete mathematics is important
This is a little difficult for physicists to understand at first, because they imagine that continuous mathematics is all that is required for physics. That's a bunch of nonsense. The real work in mathematics is in the discrete results, the continuous results are often just pale shadows of much deeper combinatorial relations.
The reason is that the continuum is defined by a limiting process, where you take some sort of discrete structure and complete it. You can take a lattice, and make it finer, or you can take the rationals and consider Dedekind cuts, or you can take decimal expansions, or Cauchy sequences, or whatever. It's always through a discrete structure which is completed.
This means that every relation on real numbers is really a relation on discrete structures which is true in the limit. For example, the solution to a differential equation
$${d^2x\over dt^2} = - x^2$$
Is really an asymptotic relation for the solutions of the following discrete approximations
$$ \Delta^2 X_n = -\epsilon x_n^2$$
The point is, of course, that many different discrete approximations give the same exact continuum object. This is called "existence of a continuum limit" in mathematics, but in statistical physics, it's called "universality".
When studying differential equations, the discrete structures are too elementary for people to remember them. But in quantum field theory, there is no continuum definition right now. We must define the quantum field theory by some sort of lattice model explicitly (this will always be true, but in the future, people will disguise the underlying discrete structure to emphasize the universal asymptotic relations, as they do for differential equations). So keep in mind the translation between continuous and asymptotic discrete results, and that the discrete results are really the more fundamental ones.
So do study, as much as possible:
- Graph theory: especially results associated with the Erdos school
- Discrete group theory: this is important too, although the advanced parts never come up.
- Combinatorics: the asymptotic results are essential.
- Probability: This is the hardest to recommend because the literature is so obfuscatory. But what can you do? You need it.
Don't study mathematics versions of things that were first developed in physics
The mathematicians did not do a good job of translating mathematics developed in physics into mathematics. So the following fields of mathematics can be ignored:
- General relativity: Read the physicists, ignore the mathematicians. They have nothing to say.
- Stochastic processes: Read the physicists, ignore the mathematicians. They don't really understand path integrals, so they have nothing to say. The usefulness of this to finance has had a deleterious effect, where the books have become purposefully obfuscated in order to disguise elementary results. All the results are in the physics literature somewhere in most useful form.
- Quantum fields: Read the physicists, especially Wilson, Polyakov, Parisi, and that generation. they really solved the problem. The mathematicians are useless. Connes-Kreimer are an exception to this rule, as is but they are bringing back to life results of Zimmermann which I don't think anybody except Zimmermann ever understood. Atiyah/Segal on topological fields is also important, and Kac might as well be a physicist.
Physics is the science of things that are dead. No logic.
There are many results in mathematics analyzing the general nature of a computation. These computations are alive, they can be as complex as you like. But physics is interested in the dead world, things that have a simple description in terms of a small computation. Things like the solar system, or a salt-crystal.
So there is no point to studying logic/computation/set-theory in physics, you won't even use it. But I think that this is short sighted, because logic is one of the most important fields of mathematics, and it is important for it's own sake. Unfortunately, the logic literature is more opaque than any other, although Wikipedia and math-overflow do help.
- Logic/computation/set-theory: You will never use it, but study it anyway.
