What to learn in pure math for applied math?

In summary, the speaker has completed their undergraduate degree in applied math and physics and is now applying to applied math PhD programs. They are interested in focusing on mathematical physics, PDEs, functional analysis, and operator theory. The speaker has been advised to self-study topics such as topology, real analysis, and numerical analysis in preparation for their graduate studies. They have also been recommended courses in areas such as complex analysis, nonlinear equations, and mathematical finance, depending on their interests. The speaker is unsure of their exact focus but is considering mathematical physics, and has been advised that some physics knowledge may be necessary for this area of study.
  • #1
creepypasta13
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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).
 
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  • #2
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.
 
  • #3
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.
 
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  • #4
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
 
  • #5
creepypasta13 said:
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

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 want to go for mathematical physics.
 

1. What is the difference between pure math and applied math?

Pure math, also known as theoretical math, involves the study of abstract concepts and their relationships, with a focus on logical and deductive reasoning. Applied math, on the other hand, uses mathematical principles to solve real-world problems and address practical issues. While pure math often serves as the foundation for applied math, the two fields have distinct approaches and applications.

2. Which areas of pure math are most relevant to applied math?

Some of the most important areas of pure math for applied math include calculus, linear algebra, differential equations, and probability theory. These fields provide the fundamental tools and concepts needed to solve problems in physics, engineering, economics, and other applied disciplines.

3. Do I need to have a strong foundation in pure math to excel in applied math?

While a solid understanding of pure math is certainly beneficial, it is not always necessary to excel in applied math. Many successful applied mathematicians come from a variety of backgrounds and may have varying levels of expertise in pure math. However, a strong foundation in pure math can provide a deeper understanding of the underlying principles and help in tackling complex problems.

4. How does learning pure math benefit my understanding of applied math?

Studying pure math can help develop critical thinking skills and provide a deeper understanding of mathematical concepts and principles. This, in turn, can aid in solving complex problems in applied math. Additionally, pure math concepts often serve as the building blocks for more advanced applied math topics.

5. Are there any resources or courses specifically focused on pure math for applied math?

Yes, there are many resources and courses available that specifically focus on the pure math concepts most relevant to applied math. These may include textbooks, online courses, and workshops, among others. It is important to do thorough research and consult with a math advisor to determine the best resources for your specific goals and needs.

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