How to make the best out of my Applied Math degree?

Question

I'm currently a second-year student taking a Math Degree. The reason I say Applied Math in the title is because I've started to see that, while I like studying Pure Math (I'm enjoying my proof based courses alot), I love to see the Math I learn be used in real life problems. So I'll most likely end up choosing electives that are "more applied" than others.

But I'm facing a very frustrating problem.

When I first enrolled in Math, I used to think that I like Math "for the sake of doing Math". However, I decided to take more Physics courses (Intro to Mechanics, Intro to E&M, a Lab course, and this semester I'm taking Modern Physics and Waves). I've also started doing lots of outside reading on Physics, and realized I'm ALSO interested in Physics.

But then I realized I'm actually not JUST interested in Physics, but really anything that has lots to do with Math and Science (mainly the more Math oriented fields). I've started reading through other topics from other fields and indeed I started to feel an urge to study them too! These topics come from a range of subjects such as Engineering and Statistics, and even some topics that aren't very Math-heavy!

I feel quite overwhelmed. I still love studying Math in all its forms, but I truly feel like I want to dive deeper into other fields. I want to learn alot.

I would like to continue into Grad school and do research. Unfortunately, I believe I must choose a certain field to concentrate on. The thing is, I know I want to study Math as a main and use that math knowledge in science and engineering.

But I feel lost. Where do I start? What exactly do I do? I need to apply to Graduate schools eventually and it won't be good to have no clear path. It doesn't seem like I want to learn one thing, but rather numerous things.

It's best to start preparing from now for Graduate schools. But what exactly do I prepare for (like, what program do I apply to)? How do I approach this problem? What advice would you give me?

Thank you in advance, any reply is greatly appreciated.

Answer 1

(Personal experience inbound in 3... 2... 1...)

One thing you have to learn is that there are way more interesting things in the world than you could possibly pursue. To me, this was most likely the hardest thing to learn to live with. There are many layers to memento mori; some things you never have found time for will haunt you, but decisions about your priorities you must make - constantly.

A typical academic is, indeed, curious, and would gladly entertain themselves thinking about topics not relevant to their own research. But, a slim fraction of geniuses - such as Hilbert, Poincaré, Feynman or Landau - aside, very few make significant contributions to science by dabbling in lots of topics professionally. On a bright side, no one is robbing you of the ability to engage in projects from different fields or having hobbies. In fact, Applied Math used to be THE place to be to work on all kinds of problems, from physics and biology to social science, until ML (arguably) took that crown in recent years. So you seem to be in a good spot!

If you are anything like me, you would also struggle with being excited by how much progress is there to be made by just thinking about it a bit and how little progress there will be in the following months doing grunt work. Alas, this is also a part of the life in academia: it is a popular adage on academia.SE that the ideas are cheap. Most of us have more ideas than time/resources to implement them.

By Robert Root-Bernstein, I would be a dilettante, but a lucky one: people with deep specialist knowledge still find my ideas and skills useful. So I still feel fulfillment, albeit the inability to stay on a more typical career track surely makes things difficult.

So, then, I believe you are facing two major options: either commit to a field and keep the rest of your interests as hobbies or commit to a skillset and try to engage them professionally. Interdisciplinary research and lack of specialization seem to become more widely accepted, so focusing on developing a skillset might be a good idea.

Finally, the question about the grad school is hard to attack. I went for something I was already working on for a couple of years but "just go for whatever" is hardly an answer given in good conscience. Grad school is not a trifle time investment, but it does not lock you out of other options for the rest of your career. Try to think of it accordingly: will it be a good time investment? What skills and connections will you likely acquire there? Be careful with something too universal though - soon as you become an expert in say fluid dynamics, military guys will want you to work for them and make your life a notch more miserable. Or not. Only you can tell.

It will be likely be hard to switch between topics rapidly, primarily because of how important networking is. But you will be able to steer your research direction for sure. If you can act upon sidetracking ideas and connect them to the right people, it would make for a great career in science. If not... Well, ideas are cheap. Best of luck.