Gaining Broad Math Knowledge: What's Left to Learn?

[Reedeegi]

I'm currently trying to gain as broad a mathematical base as possible, and here's what I've done:
Foundational:
Axiomatic ZFC Set Theory, Category Theory

Modern:
Analysis (Real, Complex, and Abstract), Algebra (Abstract, Linear), Differential Geometry, and Point-Set Topology

What are some other areas of mathematics that would be useful in gaining a very broad level of knowledge in mathematics?

 

[phreak]

Although it often fits in with differential geometry, I think algebraic topology is rather important to know. In addition, since it plays such a large part in pure mathematics, it wouldn't hurt to learn Lagrangian and Hamiltonian mechanics.

Also, it's difficult to gauge your actual level of mathematics from your description. It could be the case that you need to go a bit more in depth in certain categories to have more of a mathematical base. For example, have you studied differentiable manifolds and Riemannian geometry?

 

[Reedeegi]

Although it often fits in with differential geometry, I think algebraic topology is rather important to know. In addition, since it plays such a large part in pure mathematics, it wouldn't hurt to learn Lagrangian and Hamiltonian mechanics.

Also, it's difficult to gauge your actual level of mathematics from your description. It could be the case that you need to go a bit more in depth in certain categories to have more of a mathematical base. For example, have you studied differentiable manifolds and Riemannian geometry?

I'm well familiarized with most of the topics of all the areas I mentioned except except differential geometry, which I'm still learning. I have looked into Riemannian Geometry and I think I'll study it after Algebraic Topology. Also, would Geometric Topology be useful? Or Differential Topology? Or Mathematical Logic?

 

[Hurkyl]

In terms of elementary topics, I notice that you seem to be missing both discrete mathematics (e.g. combinatorics, graph theory) and number theory.

 

[MathematicalPhysicist]

Just don't go lunatic, that's my only advice. :-)

 

[Reedeegi]

In terms of elementary topics, I notice that you seem to be missing both discrete mathematics (e.g. combinatorics, graph theory) and number theory.

I tried studying number theory, but I lost interest rather quickly. There is something about number's I've always seemed to dislike...

 

[lurflurf]

What other have said plus

Geometry
Euclidean
Non-Euclidean
Projective
Analytic
Algebraic
Symplectic
Convex

Applied
Numerical/Applied Analysis
Asymptotic Analysis
Calculus of Variations
Finite Calculus
Difference Equations
Ordinary Differential Equations
Partial Differential Equations
Integral Equations
Integrodifferential Equations

 

[lurflurf]

I tried studying number theory, but I lost interest rather quickly. There is something about number's I've always seemed to dislike...

Have another look. Most books and courses concentrate on dull matters. Number theory is very broad and deep. It draws on many other areas of mathematics. There are transendential, computational, algebraic, analytic, elementary, and other areas.

 

[confinement]

If you are interested in the foundations of mathematics, it may also do well to look into Model theory (cf the text by Bruno Poizat) and topos theory.

 

[qntty]

I second the suggestions to study geometry and combinatorics, you've overlooked some of the most interesting branches of math!