# Advanced math subjects?

Success
After Linear Algebra, Ordinary Differential Equations, Partial Differential Equations, Complex Analysis, Numerical Methods, Numerical Methods for Partial Differential Equations, what math subject comes next? List as many math subjects as possible. But they need to be in order.

## Answers and Replies

Homework Helper
List as many math subjects as possible. But they need to be in order.
With regards to undergraduate math courses (and beyond) there is no set, linear order. Instead, there are "branches" where different courses fall. The three main branches are:

1) Analysis: Real analysis, complex analysis, ordinary diff. eq., partial diff. eq., harmonic analysis, functional analysis, ...

2) Algebra: abstract algebra, number theory, linear algebra, commutative rings/algebras, associative rings/algebras, ...

3) Geometry/topology: geometry, topology, differential geometry, algebraic topology, manifolds, ...

One could argue that logic & set theory is a fourth branch. Applied mathematics (numerical analysis, probability/statistics, operations research, game theory, ...) could be a fifth branch.

Arkuski
It honestly depends on what you want to do. As a point of reference, it is often stated that John Von Neumann (1903-1957) was the last person to know all of mathematics. It is likely that no one in the future will ever be able to have such a significant impact on mathematics as figures as Euler and Gauss had because there are simply too many subdivisions out there. As such, there does not tend to be an order for advanced subjects to be taken in, only separate sequences of subjects (Ex. real analysis then measure theory). Additionally, as the material gets more advanced, sometimes teaching yourself bits and pieces as you see fit is beneficial.

From the courses you listed, it sounds as though you might be headed in the applied direction (are you an engineer?). You should consider looking into calculus of variations as it may come in handy.

Staff Emeritus
Gold Member
After Linear Algebra, Ordinary Differential Equations, Partial Differential Equations, Complex Analysis, Numerical Methods, Numerical Methods for Partial Differential Equations, what math subject comes next? List as many math subjects as possible. But they need to be in order.

That isn't he order I did those courses.

Greger
He must mean the order in which his university offers them (in terms of prerequisites and semesters). eumyang nailed it.

aworrystudent
For physicists, usually group theory and differential geometry come next.

Success
I want to use the knowledge of physics and electrical engineering to invent new gadgets. Which math subjects are beneficial for me? Is number theory and abstract algebra important for me? How about combinatorics and harmonic analysis?

Hercuflea
As an electrical engineer you definitely want to know complex analysis and fourier/harmonic analysis. It's good that you are taking numerical methods and PDE's because these are useful for all engineering disciplines. Have you actually taken the courses in your OP yet? If you really like analysis-type math (as opposed to algebra/discrete subjects), you should also take a semester or two of Real Analysis which is a pretty classic introduction to analysis, and then move on to complex analysis, fourier/harmonic analysis, functional analysis. For functional analysis you need to have a deep understanding of linear algebra going in.