How can I effectively switch fields from physics to math post-undergrad?

Question

I recently graduated with a BA in physics. Towards the tail end of my undergrad career I took the Putnam exam (scored 28 w/o having taken any upper div math), and subsequently took a couple core math courses (one quarter of real analysis, one quarter of proof-based linear algebra). These were the probably the three most enjoyable learning experiences I've had in undergrad, and for the past month or so I've been trying to think about how I could possibly switch to a research career in mathematics but it seems very unfeasible due to the small amount of core classes I took and my inability to do any substantial math research both due to lack of knowledge and lack of faculty connections.

Right now I am taking a gap year working in a physics lab. I am technically able to audit more core math courses at my undergrad institution but wouldn't reasonably be able to do more than two (maximum three) more. I could self-study but am worried about the lack of grades to reflect my learning (although I could take the math GRE and hope for the best). It's my understanding that I'd need a strong foundation in analysis and algebra before being able to do any sort of meaningful research, so my hope is now to get into a master's program where I could fill in my gaps and start working on research, but even these programs require a much more thorough foundation than what I'd be able to build on paper. Is there any way that I could switch to math at this point, with the goal of entering a math PhD program, or is my best bet to try to do theoretical physics and segway that into mathematical physics?

Thanks in advance for your time and answers.

Answer 1

Sure; apply to a graduate program in math. Most such applicants have no publications so far, so that will not hold you back. Take the GRE. Mention your Putnam score. If you are in the US and are applying to a US school, you have a good chance of admission... not Harvard or Berkeley, perhaps, but at many good schools. You may be asked to take some make-up undergraduate course work at the start; but with your Putnam score I am thinking this will easy for you. And many math programs will let you do that after admitting you as a graduate student. Write what you wrote here in your application.

Another thought. Maybe try your own university. You have met some of the faculty already. Email the math department, see if there is a recruitment committee for new graduate students; talk to one of them and see what they say.

Answer 2

There are two things you need.

1. An excellent math GRE score. You have little to no track record, so this is absolutely vital. It's less vital for those with already good track records, although the GRE can certainly kill otherwise good applications if the score is terrible. I can't emphasize enough how important this test is, even though I think it's absolutely moronic. Start studying now. Some searching and sleuthing will reveal quite a bunch of available resources online. Free too!

2. Three excellent letters of recommendation, at least. Preferably from senior and well-known senior faculty in mathematics, who can attest to your "ability to not just survive, but succeed in graduate school". I think there's a lot of the drivel said in letters of recommendation these days, a la "strongest student I've had in the last X years". The same writer will then proceed to say that next year, and the year after that, etc. But you need this drivel regardless. (How in the world can I know if you'll be the next Grothendieck? I can't.) Don't hate the player, hate the game.

   How do I get these letters of recommendation? Great question! I've found that the best letters of recommendation come from either 1. somebody you've done a reading course with where you've produced an outstanding piece of written work for them, usually an expository paper, 2. someone you've done undergraduate research (or as others might call it, undergraduate "research") with, where you hopefully have solved some minor open problem, or 3. someone's who's (upper-division undergraduate or graduate) course you had the highest or so grade in that had competitive in-class exams (none of this take-home exam nonsense).

   (I cannot emphasize enough having at least one letter of recommendation coming from someone who satisfies my third criterion above. Unless you did your undergraduate at Harvard or Princeton and are out of this world amazing, a letter of recommendation that can say this will be vital. Indeed, many of us faculty have observed that there is a strong correlation between strong competitive exam takers and those who have a very strong grasp of the basics.)

   How can I get the attention of those elusive senior and not deadwood faculty and get them to love me enough to write me an excellent letter of recommendation, especially for my first and second criterion above? Send an email, show up to their office, catch them somewhere in the math department building, having prepared for such a meeting quite a bit to show you know your stuff. It would be a lie to say they don't bite, some certainly do. But I would say a useful elaboration of "bite" would be "is making sure they're working with someone up to their perhaps quite high standards, and that they're indeed actually learning and coming closer to attaining mastery and not just performing to impress". Sure, mathematics is ultimately about beauty, not jumping through the right hoops, but getting into a good mathematics graduate school is no easy feat these days. Once you get regular meetings going with a faculty member, whether it be for a reading course or to discuss your progress on research, you should prepare for every meeting as if you were expecting an oral examination on the relevant material.

I'll add a bit more later, if I have the time.

Answer 3

Besides applying to a master's program, which as another answer as suggested is a perfectly viable option, you could also consider a post-bac program. I believe both Smith and Brandeis have such programs. They are year-long programs and they are intended for students like you: who have a weak pure math background but are interested in pursuing a graduate degree in math. You would take most of the standard undergraduate math courses, and likely some of the lower master's courses. At Smith, I know they are also focused on bringing in frequent speakers, so you'd form strong connections for apply to a graduate program afterwards.