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I think a similar question was answered in PhD Research under guide/advisor of a different department by JeffE. I'm asking a slightly different question. Yet if you find it identical to the above one, you can report it to be 'duplicate.'

I'm interested in both pure math and machine learning, and I chose to apply to math PhD exclusively due to my background. However, recently I've been into (1) developing an algorithm related to proof verification and others for application to mathematics using the state-of-the-art deep learning techniques and (2) deep learning proper without much of sophisticated math involved. I'm certain most or all professors in math department I'm applying to (or anywhere else) aren't working in such an approach as (1), since that's more of a job of professors in CS department.

  1. Though the answer may be case-by-case, is a topic as (1) appropriate for my PhD thesis? I understand that I can have a co-adviser from CS dept., but is this topic likely to get accepted from math department?

  2. If so, by any chance if my interest totally shifts to (2) and has not much to do with mathematics at later stage of my PhD, do I still have to work on something related to math like (1)? (I assume the answer is Yes.)

  3. In fact, one of the programs I got accepted to is quite lenient in letting a math PhD student into its ML PhD program. Is going to this program the only reasonable path for the case that my interest totally shifts to (2)?

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  • Your question is, I believe, out of the scope of this forum, as it is too specific. I recommend that you edit your question to something in the form of "Is it reasonable to suggest a PhD thesis topic in an interdisciplinar area in which none of the faculty members specialize?". In which case it depends. If you are applying to the US, for instance, you don't choose your thesis topic fronthand, but rather explain in your SOP what motivates you and topics on which you would feel comfortable working.
    – FBolst
    Dec 14, 2017 at 13:42
  • Thanks for your suggestion. I'm applying to the U.S, and I understand how American programs work, other than how they accept thesis topics. I guess I will once again receive the answer "it depends", but anyway I've changed the title.
    – Math.StackExchange
    Dec 14, 2017 at 14:08
  • is [developing an algorithm related to proof verification and others for application to mathematics using the state-of-the-art deep learning techniques] appropriate for my PhD thesis? Maybe... It depends whether this topic constitutes research. can I study this topic in the maths department? Yes, if you can find a supervisor in the maths department. can I study machine learning in the maths department? Again, yes, if you can find a supervisor. (Because you can doesn't mean you should. Perhaps consider whether you should study in the maths department or computer science department.)
    – user2768
    Dec 14, 2017 at 14:24
  • Obviously, I can find a co-adviser in other department as suggested by JeffE in the above link, so my question is rather about whether the math dept. approves the topic to be appropriate or not. If you mean this question is equivalent to whether I can find a supervisor in math dept, then it makes sense. Thanks for your answer.
    – Math.StackExchange
    Dec 14, 2017 at 14:41

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Well, no answer yet? I'll give it a try, even though I'm in CS and I'm residing in Europe. But maybe it helps a bit:

To adress your questions 1-3:

  1. In CS it would - in math it depends on the will of the supervisor.

  2. Depends on your PhD program and supervisor. It might depend who is sponsoring the PhD program as well.

  3. I don't know enough about the US system to answer this one - but you asked for an other option and there is one:

  4. Just quit the program you are in and look for an other one in a CS department! Sounds a bit disruptive, but if this is your research interest, you should follow it!

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