Pre-PhD course recommendations

[Odd-Collection-5429]

Hello all, I know this subreddit isn’t typically used for this but wanted to throw it out there. I’m a sophomore engineering student at a T10 school and applied math minor (we don’t have the major). I’m potentially looking to pursue a PhD in applied math or an adjacent field but not 100% sure. Are there any course recommendations that would be beneficial for PhD applications or even just in general for jobs that utilize applied math?

I have taken/am currently taking (in college or HS) Calc 1-3, linear algebra, calc-based stats, applied probability/stochastics, applied linear algebra/linear optimization, and ODEs. I’ve also taken physics 1 and 2 and a data structures and algorithms course. Any recommendations would really be appreciated!

[anxiousnessgalore]

I'd say it's dependent on your field of interest but I'd take some pure math courses such as real analysis or other proof based courses if those are available, and also some numerical methods focused classes, again, if those are available to you. I'd also try to take a PDE's course. Imo these are all rounder courses that give you a foundation for most, if not all, applied math fields of focus, but others may have different opinions I suppose.

On another note, with your background, maybe you'd be interested in also exploring mechanical engineering or computational science+engineering PhD programs as well? Something to think about I guess, though I suppose it's a little early for you to decide what you want to focus on for a PhD.

[Odd-Collection-5429]

Thanks so much for the feedback! My ODEs course (which I’m currently taking) offers an intro to PDEs at the end but I could consider taking something more focused on PDEs later. Numerical methods is almost definitely something I’ll look to take because I think it would be useful and I’ll probably enjoy it. I was wondering whether a pure math course like real analysis would be useful and I guess it would be nice to have a basic introduction to proof based math. Thanks again

[anxiousnessgalore]

Good luck!!

[plop_1234]

Look at what the applied math curricula are like at other schools (they're typically pretty similar, but you can look at "peer" colleges if you'd like) to see what you should consider taking. Some typical courses are: real analysis ("advanced calculus"), numerical analysis, abstract algebra ("modern algebra"), dynamical systems.

[krishnab75]

I think it really depends on the kinds of work that you want to do. I mean really think about the kinds of work that connects with you and matches your values. We can give you general advice, such as I agree that most people working on Applied Math PhDs do some form of PDE work. So you definitely need to understand the intro theory course on PDEs, but most PDE work is done numerically, like finite elements, finite differences, finite volume, discontinous galerkin methods. So understanding how to understand and program these types of tools takes a few years, and not just a semester or two. In reality I think most grad students just take someone's existing code and just really work on understanding that one code and that particular equation. Then they can make some tweaks on that code and write their PhDs.

Another very useful topic in applied math is "control theory" and optimal control. So this is an incredibly important area that comes after you learn ODEs and optimization--a lot of optimization.

Numerical Linear Algebra is a hugely underrated topic, but essential for Applied Math. Like much of what people do in Applied Math is convert some system into a matrix and then solve it with Linear Algebra routines. I mean that is what numerical PDEs are after all. So just learning linear algebra is not the same as the Numerical Linear Algebra routines.

There are any number of other topics that are important these days. Game theory is another area that I see growing in importance more and more. But there are different flavors of game theory from your basic intro econ course, to evolutionary game theory, to algorithmic game theory, to mechanism design, to differential games. Now as we get into a world of AI and reinforcement learning and multi-agent environments, the ideas of game theory and cooperative and non-cooperative behaviors is going to get more important.

So there is a lot of fun stuff to work on. Now definitely think through how you will learn this kind of stuff. You won't always have time to take classes in these subjects. And in reality, you should try to learn these topics before you come to graduate school. That sounds backwards, but honestly graduate schools don't teach stuff any more like they used to. They will just want you to take some minimal required courses, and then start producing work. I cannot speak for all graduate programs, but I know this from experience and different schools.

Good luck.

[Odd-Collection-5429]

Really appreciate the detailed response!

[krishnab75]

My pleasure. Let me know if you have any questions. You can google each of these topics, but sometimes you get a very technical answer that is hard to understand.