What are the mathematical topics most essential for an applied mathematician?

Question

I am an undergraduate student of mathematics. My question regards those mathematical topics which are indispensable to applied mathematicians: what are they?

My conception is that real analysis, (advanced) calculus; linear algebra; probability and statistics; and a programming language constitute the bedrock of what he or she ought to know.

To what degree is that correct? What other topics are necessary? Would it be beneficial to study logic, set theory, etc. in-depth? What about a deep understanding of how computers work, is that needed? Or is it sufficient to be able to write intermediate-level Python programs (for instance)? What advanced topics should I direct my studies towards?

Thanks in advance.

Answer 1

General foundation: Calculus, differential equations, linear algebra. Many specific applied fields use probability (perhaps including stochastic processes and/or reliability theory), statistics (including design of experiments and regression).

Also nowadays, whether or not you consider them part of 'real mathematics', applied mathematicians often need to understand parts of computer science (perhaps including programming and database structures and/or simulation methods). I hesitate to recommend particular programming languages, because there have been so many changes over the years in which ones are popular.

It is not possible to predict everything you will need. So it is important to remain open-minded and inquisitive about parts of mathematics you don't learn during your university education.

Finally, if there are particular sciences to which you want to apply mathematics (such as economics, physics, psychology, genetics), it helps to have beginning courses in those fields.

Of course, this is an 'opinion based' question, and you may get quite different and useful ideas from others on this site.

Answer 2

It really depends on what you are going to do. When I was an undergraduate, my thought was similar to yours: I assumed that analysis, vector calculus, linear algebra, probability theory were the most important topics. However, as I am working on my graduate degree in geophysics, which is obviously not the field with the heaviest maths, I can say that I was too naive back then. To barely understand the literature in my field, besides the topics already mentioned, I needed to learn set theory, functional analysis, advanced ODE (special functions, Sturm-Liouville theory, etc.), advanced PDE, complex analysis, perturbation theory, numerical analysis, etc. And to make it clear, I really feel like I would need to understand all of these topics as deeply as I understand calculus if I want to have a chance to publish a paper!

$$$$ In conclusions, to know which topics you should go into, the best way is to read the literature published in your field. On a side note, I believe that in an era of extreme specialization, no matter which (science) fields you are going to delve into, you will need a lot more maths than just those you are having in mind!

$$$$Another side note is that, if you want to do physics, which is basically the king of all sciences, pretty much all topics of maths are needed. However, maths alone is not enough. Physics is a beast on its own. I am not saying that it is not true for other fields, but if you want to go into a hard-core physical science field (such as mine), learning physics will take you as much time as learning maths. I have seen a lot of applied mathematicians (my college professors) who work on biological modeling, ecology modeling, or even computer science, but I haven't came across many who could do physics.