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I'm a first-year pure math PhD student. I'm going to take an oral qualifying exam in a few months, and for this I need to find at least one person who agrees to be my adviser and one of the examiners in the qual. I'm interested in symplectic topology, and I have already talked with a professor of this area, whom I was going to ask for being my adviser. However, my interest in machine learning got revived, and I'm now very certain that I want to change my path.

Reapplying to CS PhD isn't really a good choice for me, since I didn't take any CS course in undergrad. However, I have studied this area by myself, and the process was very easy compared with studying pure math. I've audited topics courses in ML, from which I believe I don't have any lack of prerequisite for this subject, and that I'm ready for research once I finish reading relevant papers.

I'm aware that, once passing qual, I have freedom of changing the area to pursue. For example, a pure math PhD student having done a qual on symplectic topology can get a PhD by writing a thesis on computational complexity. While qual is not difficult for me, I have no idea how to actually proceed from now on. Should I tell the professor that I want to take the qual with him as my adviser and that I will not work on symplectic topology after that? (After that, I will find a co-adviser from CS department with whom I will work on thesis.) Or should I not tell him my intention until after the qual?

  • Where are you a pure math PhD student? In the US, taking oral exams in your first year is rare, and moreover switching fields and specialties is rather easy. In (e.g.) Europe the process is considerably more streamlined, and you may have been accepted specifically to work with your advisor. I am also a little skeptical that you actually want to switch to a field that you've never taken any courses in -- have you considered the possibility that ML seems much easier because you've taken it on as a hobby rather than a profession? – Pete L. Clark Dec 24 '16 at 23:31
  • I can't say exactly where I am, but I'm in one of the top six math PhD programs in the U.S. Since I had taken grad topics courses and seminars since sophomore undergrad, I'm ready for qual this year. I'm interested in ML not because of its relative simplicity but because of its objective. I was once interested in Voevodsky's program for its connection with proof assistant, but ML may eventually achieve more ambitious goal, such as automated theorem proving. – Weinstein Dec 24 '16 at 23:50
  • I took ML topics courses very seriously, and I've learned ML for several years. "Easy" may not be a right word. I found that reaching to the frontier was quicker for ML than, say, Fukaya category, since pure math is more structured or prerequisite-heavy than ML. – Weinstein Dec 24 '16 at 23:51
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Given that you're in the US, my advice to you is slow down. You don't need to pass your oral exams in your first year, and studying for and passing an exam in a subject that you intend to drop completely is not a good use of your (or anyone's) time. Moreover, you can stop working with (or not start working with) your advisor without jeopardizing your future in the program, so there is absolutely no reason not to be forthcoming about your concerns.

I suggest talking to your current faculty advisor and devising a program of study which allows you to explore ML while not quitting symplectic topology "cold turkey". As a first year student you can afford to spend at least a year taking courses trying to figure out what you want to study and building up background in that area. In fact, it is very unusual for a first year US math PhD student to know exactly what they want to study in the future: that kind of decision usually lies at least a year away.

It sounds like you thought you did and that you have some pride in your own advanced standing. That's understandable, but the thought of going ahead with an examination in a subject that you plan to drop soon thereafter and with an advisor that you plan to drop immediately thereafter is really not a rational, wise or kind way to proceed. I think you need to take some time to gain a better perspective. Sharing your proclivities and your plans with your peers and mentors is a good way to do so, but again: take it slow.

  • Thank you for your suggestion. I thought rapidly changing the path was necessary, since CS students tend to write paper at earlier stage than pure math students. But as you said, waiting for a year must be no problem. I will talk with my current adviser. – Weinstein Dec 25 '16 at 0:09
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This is less about process than it is about ethics.

You have knowledge about your future action that you are keeping from the professor. This is material information, in my view, that prevents the professor from reaching a fully informed decision. Thus, while he might accept to advise you in good faith, the same cannot be said for you. That is to say, you are not entering into the adviser-advisee relationship in good faith.

Let me put it to you another way. When is it best to tell my fiancee that I really don't love her but am actually only after the green card -- before the marriage or after the marriage?

Forgive the bluntness of this response, but I think a PhD is not only about the subject knowledge that you will be gaining. It's about pursuing your goals in an ethical manner. How would you feel if you were used as a means to an end? A professor agrees to take you on because he needs supervision experience to gain tenure. He really doesn't know much about the research you're conducting, something he should have said before he agreed to accept you. He achieves tenure, then asks you to leave his lab because he's decided that he really can't supervise you much.

  • Thank you for your response. I will be honest and reveal my intention. – Weinstein Dec 24 '16 at 23:30
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Reapplying to CS PhD isn't really a good choice for me, since I didn't take any CS course in undergrad.

That strikes me as a less insurmountable problem than it would be to go the other direction (from CS to pure math). Take a look at some programs of study, read the course descriptions, and chart out the sequence of prerequisites.

You have been studying Pure Math Topic A, but you are currently thinking you probably don't want to do research in it after all; if you are pretty close to being ready to take the qual in Topic A, then I think you should consider going ahead and taking it, as long as you're honest about your intentions. However, I will couple that suggestion with one other: for any Topic B that you intend to do research in, you should take the corresponding qual.

You strike me as the sort of person who would be able to get through the prereqs to start in a CS program in a reasonable amount of time -- with one caveat: I don't know anything about the subfield you're interested in, machine learning, but I do know that all students of CS must sooner or later do some programming, i.e. plan an algorithm, make a plan of attack for writing many lines of code, self-document the code, crank it out, debug it, make the input and output nice. One needs to have, or learn, a certain kind of patience. The question you wrote gives me the impression you can read and make sense of things efficiently, and do creative problem solving. I don't know if you can program. It might be a good idea to get your feet wet with that and find out if you can, sooner rather than later. For example, some people in this world take a class in CS and really struggle, because they can't get the hang of debugging. If you were such a person, then it would not make sense for you to pursue a graduate degree in CS.

Have you talked this over with your family?

  • While I didn't take the courses, I studied topics necessary for ML by myself. Programming related topics are no exception, but I cannot deny that my programming skill is currently not as good as that of CS PhD student. However, there are many statistics PhD students in this area, so I'm not worried about that. Grad students are often expected to learn things by themselves, so it is no surprise that some math and stat students learn machine learning by themselves. – Weinstein Dec 25 '16 at 6:39
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    I did not talk about it with my family, as they cannot see the difference between ML and pure math. – Weinstein Dec 25 '16 at 6:43
  • I applaud your self study approach, and I have no doubt of its good effect. // So, you've tried your hand at writing programs? Could I say you dabbled in programming? Could you add some information about the dabblings to your question? // I know communication takes effort on both parts when there's a knowledge chasm between generations (my partner, with PhD and 3 postdocs, was the first person to even go beyond high school in the family!); nevertheless, it is really worth ... – aparente001 Dec 25 '16 at 15:56
  • the effort. When a person proposes to make a sudden left turn just before arriving at an important landmark in his career path, especially. // Question: are there any other big changes going on for you currently, either in your immediate environment, or in yourself? E.g. any big changes in your sleeping habits? Please excuse the question, I don't mean to pry. – aparente001 Dec 25 '16 at 16:00

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