Maths to Study

Pure mathematics curriculum for self study with interests in foundational issues

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I wonder if I want to make my own pure mathematics curriculum to study along the next 4 or 5 years. What topics should I include? I want it to be like one which an undergraduate student of pure mathematics studies.

So, which topics are a must? Which are optional?

Now, I'm a high school student. I will leave school after a month from now. It may happen that I will join a maths department but not sure. So I want to study pure maths on my own if it's not possible to do so in University.

From my previous knowledge, I should include real analysis, point set topology, complex analysis, abstract algebra, linear algebra. Those are a must I think.

But what else?

What about, say: functional analysis, measure theory, universal algebra, lattice theory, discrete math, algebraic topology, differential equations, number theory, category theory, and geometry?

Which should I include of those? Which would be included for the undergraduate pure math student?

Until now, my knowledge of rigorous maths is only group theory, linear algebra, axiomatic set theory and mathematical logic.

I know calculus of course but not rigorous. I only know a little of how to develop calculus rigorously, mainly from Apostol's book.

My interests lie most in foundational issues, so set theory and logic are musts!

Any advice?

Thank you for your time :)









Recommending what you should study is easy; in fact, you have covered essentially everything you absolutely need before a PhD program. And I can't think of anything to add (except maybe differential topology which was a senior undergraduate course for me).

However, recommending what in which order is an entirely different matter. And this is the more interesting question.

If you were a computer that could comprehend perfectly the things you read, I would tell you to study these things in order:

Logic
Set Theory (Set-class Theory)
The Natural Numbers
Category Theory
Order Theory
Group Theory
The Integers and Number Theory
Ring Theory
The Rationals
Field Theory
Point-set Topology
The Real Numbers
The Complex Numbers
Linear Algebra
Measure Theory
Real Analysis
Complex Analysis
Functional Analysis
Differential Equations
And I would tell you to study Algebraic Topology anywhere after Group Theory, Point-set Topology, and the Real Numbers at your discretion. And I'd recommend Geometry anywhere after Set Theory (preferably after Group theory).

However you're not a machine, and you're not going to master these subjects in this order. Undergraduate curricula do not resemble this list in the least. And if there was a piece of advice I could give you that I wish someone would have told to me before I decided to stick in mathematics, it's that mathematics is a gigantic kaleidoscopic marsh. This is because different branches of mathematics often intertwine (especially true in Abstract Algebra, Number Theory, and Linear Algebra), reach back and support each other (such as Topology and Real Analysis), play off of each other (such as Algebraic Topology), or fundamentally change the game (such as Set Theory). To fully chart one piece of this marsh will require several passes over the same area in different directions.

But even though these branches wind around each other, they each offer their own unique tools and ways of thinking that enable you to traverse the marsh in different ways. This aspect of mathematics can be frustrating as it forces you to sit down and re-wire your brain to think in a different way. This is quite difficult, and, broadly speaking, most people settle into an 'algebraic' or an 'analytic' pattern of thinking.

Since you are only leaving high school without much knowledge of where you're going, you are most likely at a disadvantage. Something should be said about being forced to do homework, take quizzes, and exams that require you to sit down and do the work (unless of course, you are a highly motivated self-learner) so that you are forced to exercise your brain in these different patterns.

But it is well worth it; after this, you can recreationally combine distinct structures from two or more different branches to consider a new compound structure and wonder if the structures interact non-trivially (such as topological groups, measurable topological spaces, or topological vector spaces) which, in turn, leads to new questions very worthy of research.

But you're mostly interested in foundations. This is mostly associated with Set(-class) Theory and Logic. However, you should seriously look into Category Theory. Category Theory is fundamentally different from Set Theory in that Set theory cares more about 'things', but Category Theory cares about 'the stuff between things.' But most interestingly, like Set-class Theory can be used to model Category Theory, Category Theory can be used to model Set-class Theory (namely in the theory of topoi). It's an interesting duality.

I hope my short insights help you in your study.

(Learning what from whence is also an interesting question, but there are already many questions about books on this website. One piece of advice though: no mathematician was ever harmed by having too many books)