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I am applying to pure math phd programs, and looking to study algebraic geometry. I have one very strong letter from a professor who supervised some algebraic geometry research I did. I am not sure about the other two letters. I think I may be able to get a decent letter from a math professor who taught two courses I did well in. I was not on campus very much during the past year so I do not think there is a third math professor who even knows me well enough to write a good letter.

I worked in a physics lab as an underclassman for several semesters. The PI thought very highly of my work (and said so) and we had a good relationship. (Unfortunately COVID prevented me from doing any further work in his lab.) Would it be wiser to get a third LOR from this person or from a math professor who barely knows me?

I think such a letter would showcase my capability to do research in general, and my willingness to solve problems outside of "my own" field. However, the PI wouldn't be able to comment on my potential in pure math, so I'm concerned his letter might simply be out of place. Also, although I had a good experience working in his lab, I haven't spoken to the PI for a year now, and while I'm sure he would write me a letter, he wouldn't be able to talk about anything I've done recently. Hence the letter may come off as "out-dated". Finally, I don't actually want to pursue research in physics in the future, and I don't want math phd programs to get the wrong idea that I am unsure of my career path.

The other option for my third letter writer is some professor in the math department (I'd basically be picking at random at this point) who doesn't know me personally but taught me one course. I understand neither option is ideal, but is it better to get a letter from a professor who knows me personally but which is out-dated and off-topic, or from a professor who doesn't know me personally but which is current and on-topic?

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    I would argue that a professor who "doesn't know you personally" is not a very good choice for a letter of recommendation, particularly when you'd almost be picking at random. Jul 16 at 16:57
  • Was the undergraduate research you did with the physics professor experimental, computational, or theoretical in nature? Jul 19 at 16:27
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To add on to Buffy's answer, it's not uncommon for someone to do research on a different area in undergrad than they do in their graduate studies. Any research experience will be of benefit for a PhD program, even if it is not directly related to what you want to focus on for your PhD.

Physics and pure mathematics, at least from my undergraduate experience, are very closely related. I'd get the physics prof's recommendation, especially if they could speak more highly of you than a math prof who doesn't really know you. You say you haven't spoken to that physics professor in over a year, and that you are concerned about his ability to speak of you as a math researcher. Before or when you ask them for their recommendation, this is something that I would discuss with the professor.

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I'll assume this is for the US. It might possibly be different in other places with a lesser importance given to such letters.

A person who doesn't really know you won't do your application any good. What can they say that is truthful and predicts your future success and knows whether you work hard or coast along.

Even a letter from someone who can only say "this person did well in my course" isn't of much help.

Since the undergraduate program in the US is very general, it isn't uncommon for people outside your major to know things of value, especially if you have worked closely with them.

The letter from a physicist that you have worked with is almost certainly better then from a mathematician who doesn't know you - assuming of course that they praise you honestly.

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