What to learn in pure math for applied math?

creepypasta13

So I finished my undergrad last year in applied math and physics. I'm currently applying to applied math phD programs (but they are separate depts from the pure math depts). I don't exactly what I want to focus on, but I was thinking something in mathematical physics, PDEs, functional analysis, and operator theory. Perhaps the program I go to will let me work with a pure math prof doing stuff in string theory

The applied math courses I've taken include proof-based fourier analysis, linear algebra, and analysis. Also, courses in prob/stats, complex analysis, ODEs, PDEs, dynamical systems, and numerical analysis. So what should I self-study in the meantime? I was thinking topology or the second half of real analysis (integration, metric spaces. Lebesgue, etc).

Chaostamer

I've been told by three advisor-type people in my department that analysis is absolutely necessary for any math program (and as such, all math majors are required to take one semester). Since most of the math grad programs I've looked at start with a year of analysis study, I'd recommend doing as much of that as possible. Topology is probably a good idea too.

Msh1

It really depends on the area of applied mathematics that you want to work on and taking certain courses will be completely useless in other areas, for example if you want to study string theory then a course such as algebraic topology or non commutative geometry seems good but that has almost no applicability in most other areas. However, there are courses that let you keep your options open. I would recommend any of the following courses, if you have not decided on your specialty yet.

definitely study complex analysis if you have not taken a course in it already.
a second course in partial differential equations
a course in applied nonlinear equations
As many courses as you can in numerical analysis( a good choice is computational methods for PDE's or high-performance scientific computation)
a course in linear programming
a course in combinatorics
maybe a course in control theory

If you are more into mathematical physics then you can take the following courses that don't require serious knowledge of physics.

Differential Geometry
mathematics of Fluid Mechanics
mathematics of Quantum Mechanics
mathematics of Quantum Field Theory
mathematics of General Relativity

If you are interested in theoretical computer science(which is a branch of applied math) you can study,

Computational Complexity Theory,
Advanced Algorithms Design
Automata Theory
Cryptography (cool course! )
Mathematical Logic
Category Theory
Set Theory

If you are interested in mathematical finance:

As many courses in real analysis as possible
As many courses in statistics, probability.
a second course in numerical analysis.
a course in nonlinear optimization
a course in mathematical theory of finance.

BTW, take topology only if you are going into mathematical physics, or you want to do serious
coursework in real analysis, other than that topology has little applicability in other areas.

creepypasta13

I don't exactly what I want to focus on, but I was thinking something in mathematical physics, PDEs, functional analysis, and operator theory.

If I wanted to do the math of QFT and relativity, I sure hope those don't require much knowledge of physics. I hate studying relativity

Msh1

First of all, I don't think you should go for mathematical physics if you hate studying relativity. After all, all those courses do involve physics.
But I am pretty sure that the courses I listed under mathematical physics don't require any serious knowledge of physics, I myself took General Relativity and did well. The only physics courses I had taken were general physics I and II. The only course requirement for that was introductory differential geometry. Mathematics of QM and QFT require some knowledge of PDE's operator theory and functional analysis and basic probability and again no physics beyond freshman year. Topology is also very helpful in QFT and latter on if you want to study a specialized course in string theory. So I think overall topology is a good idea if you wanna go for mathematical physics.