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?