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My question is specific to admissions in the graduate mathematics. I have had an unconventional training this far; I started off as an Economics major. I realized that I enjoy doing physics and mathematics much more, especially when I had my first exposure to a proof based course in mathematics. I then started taking physics and mathematics. Unfortunately, though, I took more physics than mathematics courses, even though I seem to be more interested in mathematics.

I am contemplating applying to graduate school in mathematics, hoping to work under the supervision of someone who works in mathematical physics. Most mathematics departments mention that 3 letters of recommendations are required. Here's my situation:

I can arrange one mathematics professor, say Y, to write a letter for me. I have taken a second course in analysis with her, and I have been her TA for the first course in analysis twice. Other than that, due to my unconventional background, I have been studying mathematics with her out of class, and I continue to do so during the next year which I shall be taking off. My other two letter writers will be physicists: one works in the field of quantum dynamics; by the time I graduate, I would have had 4 courses and a paper with him, though I don't see myself working in his area of inquiry. The other one will be, hopefully, from a high energy physicist, with whom I took 2 400 level physics courses, and studied a bit of QFT in the summer. I'm not sure if I will continue working with starting summer as he is due to leave for a sabbatical.

Should I arrange for more letter writers who are mathematicians? If so, what difference would that make in my case? For instance, suppose I arrange letter write X. With X, the best I can do in the next few months is to cover some material that I haven't done in a course. I can do the same material with professor Y as well, who seems open to the idea of my discussing the material I self study with her. Or, should I contact another instructor, X, and perhaps try and do something else with him or her. If so, what options should one choose to strength the application?

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It is certainly ok for your letter writers to be in an area that is different from your focus for your PhD. Based on my experience, this is more the norm than the exception.

The letters should describe why you will be successful, not something about particular research fields. Having strong letters from your physics profs will help you far more than a weak letter from a math prof that barely knows you.

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    Seconding this advice: in most cases, physicists can accurately appraise your analytical/mathematical abilities and interests. Absolutely go with people who know you well, and can comment in detail on your potential in those directions, rather than people who barely know you but have the "right" title/label.
    – paul garrett
    Apr 13 '17 at 23:48
  • experience? you are a grad student
    – Rüdiger
    Apr 14 '17 at 0:01
  • @Rüdiger Yes, I've known a lot of people that went into PhD programs that differed considerably from their undergrad field.
    – Austin Henley
    Apr 14 '17 at 0:03
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    @Rüdiger, whatever Austin H's limited experience, his opinion on this is accurate. As "an old person", I've been Dir of Grad Studies in a pretty-darn-good math dept on two different occasions, and on the grad admissions cte for maybe 30+ years, etc., and "can confirm". True, one should not necessarily trust opinions of people who lack experience, but, ... whatever... in this case "I endorse it", for what that's worth.
    – paul garrett
    Apr 14 '17 at 0:16
  • Good answer. However, sometimes a letter falls through, e.g. a professor is out on a medical leave. Do cultivate the possible fourth reference. You can decide later.
    – aparente001
    Apr 14 '17 at 2:20

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