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A few years back, I did an undergrad double major in math and physics, and afterwards went to a top-tier university to work on a physics Ph.D. Around about my third year of the Ph.D., I started to feel like my research was not really going anywhere, and I had serious thoughts of dropping out. I sought out advice though, and the word I got was always "Everybody feels that way in their third year. Stick it out and it will get better."

Unfortunately, it did not get better. I'm now in my sixth year of the Ph.D., and in all the time I've been here, my entire research group has only published one paper. It may take another year or two before I get to be lead author on a paper, and even then, I don't think the paper will be very good. Moreover, I have little desire to do this work for the rest of my life. And so I am again thinking seriously about what alternative options I have.

While I've published no physics papers in my time at grad school, I did publish two small notes of original research in Amer. Math. Monthly on my own, and even though they're not terribly consequential papers, I'm happier about them than about my "official" research. I enjoy math research, but I don't think I know enough to do it professionally (my physics research is not something like gauge theory that is of direct interest to mathematicians). I've been considering dropping my current program and starting over in a math Ph.D. program, but it's not clear to me whether this is a viable option.

So my questions are whether good math Ph.D. programs would be willing to take a person like myself? How damaging to my credibility as a researcher would it be to drop out of a Ph.D. program? Should I even mention that I was once a physics Ph.D. student, or just leave a big blank spot in my resume for those years? And finally, if this course of action is realistic, how should I go about getting recommendations (I would not expect my advisor to be very happy if I dropped out, and I have no real connections in the math community at this point)?

Obviously, I'm not going to do this without careful consideration of many factors, but I wanted to solicit the view of people in math academia on whether I'm being at all reasonable, or if I'm just a crank.

(I can provide some more situational details if it is necessary, but I would like to maintain anonymity and genericity to the extent possible.)

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    $\begingroup$ I personally know two highly successful full professors in top-20 departments, one of whom spent several years driving a cab before attending grad school and the other of whom worked as a janitor. I can't imagine that six years in a physics PhD program would be poorer preparation than those people had. $\endgroup$
    – Steven Landsburg
    Commented Oct 30, 2020 at 0:04
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    $\begingroup$ You might get more answers at Academia Stack Exchange. $\endgroup$
    – Ray Butterworth
    Commented Oct 30, 2020 at 0:32
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    $\begingroup$ Doesn't your university have a math department with professors from whom you have taken classes? They might give you more informed advice than a forum on the internet where people know neither you nor your work. $\endgroup$
    – Michael Renardy
    Commented Oct 30, 2020 at 1:26
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    $\begingroup$ It can't hurt to try applying to math Ph.D. programs. But don't leave a blank spot. It just invites people to use their imagination to fill in that blank spot with the worst possible things that they can think of. $\endgroup$
    – Timothy Chow
    Commented Oct 30, 2020 at 4:30
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    $\begingroup$ @Andrew: I don't think a link to a paper would be compatible with the OP's (very understandable) wish to remain anonymous. $\endgroup$
    – Jochen Glueck
    Commented Oct 30, 2020 at 8:47

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Certainly it is possible to "jump ship" and, as the comments have already pointed out, a few years in a physics PhD program (where you published math papers) will not damage your credibility. In fact, I'd encourage you to market it as a strength. When you apply to math grad school, the fact is that you already have math publications (most other applicants don't). You already know how to do research. You already know how to pass qualifying exams and give talks. You are a lot stronger than the other applicants. In terms of letters of recommendation, I'd pick your favorite professor and ask that person. I'm sure they care about your well-being in life and would not want you to do a career you hate. This professor can speak to your strengths as a student, your strengths as a researcher, your passion for math, and how you got along with people in the department. That's plenty enough to get into a PhD program in math. The letter doesn't have to be from your physics PhD advisor.

I'd also market your time in physics as a strength when you do eventually apply for jobs after finishing the math PhD. I've been on about 10 hiring committees and can remember several cases where people had jumped ship, including a few where the person had actually already finished their first PhD then jumped ship and gotten a second PhD. We were really impressed with these candidates, and I remember one case where the person got a job offer from another university before we could even schedule an on-campus interview.

So, instead of saying "I hated physics and quit" you want to say something to the effect of "I have experience doing research in both mathematics and physics. This enables me to do interdisciplinary research, and to create collaborations with professors both in and outside of your department. It also informs my teaching, as I can better connect with math students who favor a physical intuition. And, it informs my advising of students: how to make decisions based on their passion, how to trust their instincts, and how to be resilient."

Now, I want to be clear that I'm at a liberal arts college, where teaching and advising are super important, and where we like candidates with a breadth of knowledge and the possibility to do interdisciplinary research. Even if your math research has nothing to do with physics, it's a benefit that you have such a deep knowledge of physics. It could help, for example, in teaching math courses aimed at students trying to do engineering or physics grad school (like, div, grad, curl for physicists, heat equation, etc). These days even big research universities care about your teaching, so I don't think the physics time would hurt you.

One last point. You've already sunk six years into the physics PhD. All else being equal, it would be better to have finished it before starting the math PhD (which, by the way, you can probably do a lot faster than six years with all you've learned about research) rather than to leave with no degree. If you think you can finish it within a year or so, I'd encourage you to do that, even if you plan to jump ship.

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    $\begingroup$ (+1) (and more if I could) for your 3rd and 4th paragraphs. Of course, the OP shouldn't make it as "in your face" explicit as this, but what you've suggested shows the kinds of things the OP may later (when finished with math Ph.D.) want their application materials to advocate for and provide evidence of. And these things certainly would be useful for getting into a math graduate program. $\endgroup$
    – Dave L Renfro
    Commented Oct 30, 2020 at 15:58
  • $\begingroup$ I want to second the advice to finish the PhD (if that is feasible). First, you may find yourself needing a job at some point -- life is unpredictable -- and a PhD is far more useful for this purpose than an ABD. $\endgroup$
    – user1504
    Commented Oct 30, 2020 at 19:10
  • $\begingroup$ Second, I'm not sure a math PhD is required to do math research. The math community really is a community (and so much of its lore is oral). To take part in math research, especially in a paid capacity, means to join this community. A PhD is one way to achieve this. But so is a postdoc or a visiting position. The key thing is doing work and talking to people. Start now, if you're going down this road. $\endgroup$
    – user1504
    Commented Oct 30, 2020 at 19:11
  • $\begingroup$ @user1504: I'd agree that the paper qualification isn't needed, but what one does need is the ability to find and/or create problems, discern what problems one can fruitfully work on (and what problems should be set aside), solve them, write up the solutions well, and get others to recognize the value of what one has done. For most people, going through the process of getting a math PhD under a good advisor provides invaluable training in these matters. The OP has some research experience, but it sounds to me that he or she hasn't quite reached the point of being able to function as a postdoc. $\endgroup$
    – Timothy Chow
    Commented Oct 30, 2020 at 23:27

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