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I am an undergrad from a pretty unknown Asian university and I would be applying to PhD programs (mostly US) in pure mathematics this application cycle. However, my major is physics and my intention to switch fields came relatively late, and therefore very few math professors know me very well. I have taken as much upper division pure math courses as my schedule allowed, with courses in analysis, algebra and geometry/topology. I will hopefully have taken 11 one-semester upper division and graduate courses by graduation, but 6 of them will be taken this year.

I am now trying to gather the three recommenders needed for application. My thesis advisor is a math professor and I have been doing some independent study under him for quite some time, but I was struggling in choosing the remaining recommenders. I have a few candidates:

  1. A physics professor from an Ivy League university supervising a summer research project on supersymmetry, though this particular project is not very mathematically intensive,
  2. A math professor from my home uni teaching me a graduate topology course this semester (this is the first graduate course that I have formally taken due to curriculum constraints); I don't have much interactions with him though,
  3. A math professor from my home university who taught me an undergrad algebra course on representation theory last semester: we had a bit more interactions but I haven't worked closely with him,
  4. A math professor from my home university that I have been doing some small group study under, but he is not very closely involved (our small group does meet weekly but he only occasionally comes in and discusses) and we started fairly recently; since I also hope to do an MPhil under him if I don't get an offer, I'm a bit worried whether asking him for a letter at this stage will make him feel negatively about taking me in as a master student.

Any advice would be helpful. Thank you!

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You want the letters that can say the most about you beyond how you did in a class (that's in your record) even if their expertise is not the closest to the mathematics you want to do in grad school.

That suggests your thesis advisor, and #1.

I would hope that your ambition to go for the math PhD would appeal to #4 even though acceptance would mean they wouldn't have you for a master's student.

Related: Postdoc Letter of recommendation same school or different school

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Broadly speaking selection for PhD's these days is fundamentally premised on completion risk: a good risk is someone who looks like they are:

  • Eligible to apply;
  • Adequately prepared* for the demands of the program;
  • In the case of research degrees - able to complete the research proposed given the time and resources available;
  • Likely to produce quality* work (to a standard at least sufficient to pass, but at best to make them look good);
  • Likely to complete on time.

*Noting what 'quality' and 'preparation' can mean something quite specific depending on the discipline area.

They are likely to use all of the evidence they gather from you during the application process to inform their decisions against criteria that will - either explicitly or implicitly - look something like the list outlined above.

Most application processes require several referees or letters of recommendation.

With the above in mind, one strategy would be to consider different aspects of the proposed research and what makes you a good choice. For example, you may have a great referee on content knowledge but another one on method or skill or innovation or the like and another that may be more about working style or approach to problem solving.

Having said all that, the closer the referee is to their own department the more sway it is likely to have, and everybody likes an Ivy League branded referee - especially if you have worked directly with them and they can speak to the prospect that you would be a safe bet.

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