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I am a sophomore at a competitive university, and I have been interested in mathematics for most of my life. As a result, I decided to take up math in university. I am ahead of most other students when it comes to math, having already taken a graduate-level course in differential topology. I have been told I have what it takes to be a mathematics professor. However, I am reluctant to pursue a pure math PhD. I am getting to math courses that are very interesting, yet are not needed for most careers (like advanced number theory). Though mathematics is my greatest strength and an interesting subject, I am interested in a variety of other subjects.

I have poured a lot of time into studying math. Halfway through my studies in the university, I wonder whether I should spend a significant amount of time studying some other subject in depth with the hope to get a career related to that subject. So, my question is: should I keep studying math at my current intensity, even though I have mastered basic math courses (algebra/analysis/topology), or should I dial back my intensity in math and spend significant time to get a deep understanding in a different area?

Of course, I have taken introductory courses in other areas, but there is no one area besides math I have studied in depth. So, this is a significant decision. I know this question is subjective, thus I only ask for the pros and cons of both paths. Two additional subjects I am interested in are computer science and chemistry/biochemistry. I feel fortunate to have a good understanding of basic math this early, and I repeatedly get asked what I will do with this math, and I am divided on the answer. I am specifically interested if anybody has any thoughts on the combination of chemistry and math.

I also encourage anyone to share if they have suggestions for areas outside of STEM that they think a mathematician would benefit from understanding deeply. I was actually raised in Washington DC which provoked activism for certain political issues. My point: I am not against studying humanities as a mathematician. My good friend (a few years older) has done the reverse of this: he went to college mostly for humanities and currently attends graduate school for math, even though he knew he had an interest in math from high school. Any advice on these questions is welcome, and thanks in advance to anyone who takes time to provide advice.

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    "courses that are very interesting, yet are not needed for most careers". I don't think that this is how studying mathematics feeds through into the many careers where mathematicians excel: imho it's not the content that matters, it's a whole way of thinking that is being developed. May 5, 2023 at 11:25
  • I think you might be interested in the advise given by Sir Roger Penrose (whom I'm a great fan of) in this talk at Churn institute (watch starting 1:57:00): youtube.com/watch?v=25HjUmn0KCE. My personal experience seems to be in accordance with what he says. May 5, 2023 at 15:57
  • Word of warning: if you do go to grad school in math, you're likely going to have to repeat course work if you take too much advanced course work in math in your undergrad. You typically can't double dip and transfer those credits in, so to speak. Focus on research instead if you decide you want to pursue math. May 5, 2023 at 23:52
  • You study whatever you like to learn. Why bother to ask?
    – Nobody
    May 7, 2023 at 9:28

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Based on what you say here, it seems you have never done any mathematics research. The job of a working mathematician is very different to studying undergrad or even graduate level math. I don't believe that people can have judged that you have what it takes to become a professor without you having significant research experience.

That said if you feel reluctant to do a pure math PhD then you should (in my opinion) emphatically not do one. A PhD is a long and hard process and if you don't enjoy it for its own sake it is generally a terrible idea.

One thing that is important for you to understand is that most people don't end up working in a career which is directly part of, or even related to, their undergrad degree. A lot of math graduates go on to become programmers, or accountants, or data scientists, or physicists, or inventory managers, or a hundred other jobs that neither you nor I have heard of. Studying pure maths at university does not mean that you will study pure maths for the rest of your life.

Combining chemistry and math, or computer science and math are both fine ideas. I know multiple (UK) universities which offer chemistry and maths, or computer science and maths joint honours degrees (essentially doing half a degree in each subject). If you are interested in studying these subject then you should, but don't feel obliged to for some hypothetical future career.

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  • there I thought physicists who couldn’t find a physics job became mathematicians…. I don’t think people with math degrees “become” accountants either. They might do accounting or physics degrees, but then they function as “properly” trained professionals in their fields. May 6, 2023 at 19:50
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Most of the advice here is about what to do for grad school. I say start simpler by giving yourself a bit of breadth if you think you don’t want to stick with math (you don’t have to stick with something just because you are good at it).

From your comments it seems like you might be interested in things with a bit more of a human application. Check out economics, bioinformatics, data science, network science. Browse some textbooks and have conversations with faculty. Enroll in some mid level courses but don’t feel you have to commit until you try a few things. Good luck.

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Most of the following is not necessarily an answer nor "advice," but rather a collection of anecdotes.

I knew fairly early on during my undergrad studies that I was going to go to graduate school and do a PhD. I didn't know at the time exactly what I would do with the PhD but I had a good feeling that the kind of careers I was interested in would require that advanced knowledge. In my undergrad, I majored in Pure Mathematics and Combinatorics & Optimization. I wanted to get a minor in Statistics as well because I was thinking: ok, statistics, quantitative finance, those seem like solid career options. In the end, I didn't end up doing the minor because of scheduling conflicts.

Now in grad school, I entered thinking I might end up doing a postdoc and hopefully, eventually becoming a professor. I still took some probability theory courses because it's always good to have options. Otherwise, I started my research, working in combinatorial algebraic geometry and things were ok, I was never a star PhD researcher but my research was "adequate."

Then the COVID-19 pandemic hits and my mental health and my research hit a slump. This began midway through my 3rd year of my PhD (which is a 5-6 year program) and about the time I needed to start thinking about my career after graduating. I realized that research wasn't exciting for me. I still liked it, sure, but it didn't rouse me the same way my undergraduate courses did. But I would have never known this before having started my PhD.

At the same time, I found out that my coauthor was leaving academia to work in industry. This coauthor's research was more pure than mine and he had done 2 postdocs with intentions of going into academia; didn't have much statistics/data science training until near the end of his last postdoc. He's now working as a data scientist. My takeaway is that it is never to late to make a transition from pure math to industry. Of course, all things being equal, it is preferable to have the industry training earlier than later but also don't give into the sunk-cost fallacy and think it's too late to make that transition.

After the pandemic "ended," I took a couple graduate statistics courses because I still wasn't 100% sure which path I would take after grad school. When I started applying for jobs last Fall, I was thinking about what excited me and that was outreach and teaching but outreach doesn't have as many options for a career so I focused on teaching. Also I've heard that tenure-track faculty positions can have 100 or more people competing for 1 position (yikes!). I felt that I wanted to stay in academia for the culture but not so much in research. So I was looking at teaching-focused positions or some government positions in statistics (similar culture, I felt) and a few postdocs because it didn't cost much to apply. I was recently accepted into a teaching-based position with an equivalent of a tenure-track and I'm quite happy about where I landed in the end.

In my time here, I ended up hanging out with several people who worked in teaching-focused positions (at my school) or graduated and went on to such positions. Maybe it's a case of like-personalities attract. Some other anecdotes of people I've known/met:

  • math undergrad, now doing CS for PhD
  • PhD in (pure) algebraic geometry, now working in AI
  • PhD in algebraic number theory, going to work in a government lab (I believe in quantum computing)
  • Exited PhD with a masters degree, going to work in a government institution
  • PhD in combinatorics, looking for work in quantitative finance or maybe data science

My school maintains a list of where PhD alumni end up (ACO = Algorithms, Combinatorics, Optimization; CSE = Computational Sciences and Engineering). Although I acknowledge, it doesn't fully specify what they studied in grad school.

There are also postgraduate programs which are not "math PhD/masters" that you can consider. Law schools take people with any bachelors degree although I don't get the sense that that's an easy route to take. There are also math education PhD programs. Even looking at "math PhD" programs, many school offer different flavours like math-bio, quant. finance, combinatorics/optimization, computational math. Or you could apply to a CS program. I haven't read it myself, but see if your library has the book "101 Careers in Mathematics."

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  1. You may consider applying for a Masters/PhD programme in an area different from Pure Math. Needless to say, you will be welcome at an Applied Math or CS department. Above that, your application will be gladly considered by professors working in, e.g. quantum field theory (QFT), superstrings, quantum gravity (and gravitational physics in general), celestial mechanics, orbital mechanics. Many schools in the US will then enable you to take remedial undergraduate courses during your Masters/PhD studies.

If you choose to shift to theoretical physics, it will be advisable to read up some quantum mechanics at an undergraduate level. Normally, I would recommend Griffiths. However, given your mathematical education, you better use this little masterpiece. It will also be good to make some acquaintance with Relativity. To start right away with General Relativity concepts, the online course by Sean Carroll may be used. The best textbook on Special Relativity ever written is the little-known text by Stepanov. Spend 60 bucks on it, and you will be in for a treat.

If you decide to consider celestial and orbital mechanics, try to read this masterfully written introduction. It will help you understand if this choice is good for you.

  1. Regarding humanities. Beware that in those areas you will encounter a subculture different from that of STEM. While in STEM people still are judged mainly by merit, in humanities (especially in the so-called ``critical studies'') one's political agenda becomes a key factor in their career. If you are comfortable with tuning your research to the ever-changing party line, and if self-censorship doesn't scare you, you may give it a try.
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  • I downvoted. If you have a background in math and want to switch to physics, the last thing you need are texts presenting mathematical approaches to QM or GR: such texts are useful if you want to continue in math but specialize in QM or GR. Physicists do not think like mathematicians and any candidate physicist will need to well to learn the lingo of physics if they are to be understood by that community. I'm not a great fan of Griffiths personally but it is a classic in its exposition and typical of the physicist approach to QM. May 6, 2023 at 20:24
  • @ZeroTheHero Physicists' texts open with the Schr eqn -- but fail to explain whence it emerges. They do not derive it but introduce it out of nowhere. They fail to show that the duality between the observable and density operators is analogous to that between the classical observable and distribution. They fail to explain that the Schr & Heis pictures in QM are analogues to the Liouville & Ham pics in stat physics. They never stress that an expansion of a density op over projectors is analogous to the expansion of a distribution over classical states with both the momentum and position fixed. May 6, 2023 at 21:16
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I think it's a good idea to study something else in addition to pure math. It's a good idea to keep your options open. I think it would be a good idea to explore different career paths before deciding to do a PhD in math. (Actually, you have to get in first, and I know people who applied to PhD programs and didn't get in.) Don't apply to PhD programs because you don't know what else you would be doing. Also, it would be a good idea to learn about what life as a professor is actually like. I used to think that mathematicians just spend all day thinking about math, and that is very, very false.

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