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I did research and advanced in the proposal, but suddenly my advisor left the university. They said that I must choose a new adviser. I know some general criteria about selecting an advisor. could you please mention some specific things that I concern in order to reach to best case. for example, his/her scientific background is important yet?

P.S: It differs from the following question, that is about general case, but in our question, assumed that we have a "semi-solved research".

Why would one choose a particular advisor, other than having shared interests?

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    This question makes it sound like your advisor simply vanished, without helping to set up transition plans for his/her students. Under ordinary circumstances that would be highly unprofessional, and even unethical. (It might be excusable in some cases, such as mortal illness.) Assuming you really aren't getting any help from your former advisor, understanding why could be a good first step. Leaving abruptly is itself unusual, so something strange is going on here. – Anonymous Mathematician Aug 10 '15 at 11:41
  • @AnonymousMathematician do you have any suggestions that I ask him? He only can't travel and the faculty didn't accept his conditions! He is ready for any help. – moksef Aug 10 '15 at 11:48
  • That's good to know. A key question is whether he has any recommendations for new advisors who might be able to supervise your work. For example, collaborators of his who know the subject well, or people he feels might be helpful/accommodating in this difficult situation. You could also bounce ideas off him ("X, Y, and Z occur to me as possibilities, with the following advantages and disadvantages. Do you have any thoughts on the matter or recommendations for how I might choose between them?"). – Anonymous Mathematician Aug 10 '15 at 13:24
  • @AnonymousMathematician Thank you for the point. – moksef Aug 10 '15 at 13:29
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I feel this is a difficult situation, with a number of variables that should influence your decision:

  1. Is your advisor completely gone (e.g., in complete retirement, gone to industry, or even dead), or has he just left for a different academic post? If he is completely gone, you may need to find somebody to supervise you in your day-to-day research. If he has just moved on to a different university, it is possible that you want somebody to "formally" be your PhD advisor, but practically keep on working remotely with your old advisor. Whether you even want to keep on working remotely with your old advisor depends on how advanced you are and how well you worked with her/him beforehand.
  2. Are you advanced enough that you can finish your thesis more or less on your own? That is, do you need an advisor or an examiner for your thesis? If it is the latter, then pretty much anybody who is sufficiently well-versed in your area could act as your advisor for the finishing sprint. If you actually feel like you still need a lot of input to get your thesis ready, it will be harder to find somebody who can and is willing to take you on midway. If you are advanced and your advisor is only moving to a different university, it should be reasonably easy for you to find a "local proxy" without having to adapt your research plan too much.
  3. How set-in-stone is your research plan? Or, in other words, how ready are you to adapt to the research agenda of another advisor? Generally, if you really don't want to change your research, but you also feel like you still need a lot of input, it is likely that you will have problems finding a suitable new advisor. If you are happy to change research topics, finding an advisor should not be more difficult than the first time.
  4. How does your funding work? That is, do you "only" need a new advisor, or do you need a new advisor and a new funding source (e.g., project, grant)? Clearly, the former case will make it much easier to find a new advisor. In the second case, you will need to be willing to change your research plan somewhat, as few professors will be ready to dish out grant money if the student is not actually working on their projects.
  5. Are you a good student? Ok, this may be somewhat controversial, but clearly if you have been an overperformer so far, it will be substantially easier to convince another professor to take you on. Similarly, if you already have great publications in your research, it will be much easier to convince a professor that you want to continue working on this concrete research idea and not change to whatever other project your new advisor has in mind.
  6. Are you willing to change universities? If so, maybe coming along with your old advisor is an option?

So how does this help you? In a first step, you need to reflect on all of these points to figure out what kind of advisor you actually want/need. Then go over the list of potential faculty that formally could serve as advisor. Think about how your collaboration with each of those potential advisors could work, and what you would need from them. Get in touch with the ones that seem most suitable (as you already study there I am assuming you can just send them a mail, or have your current advisor put you in contact), and discuss the issue with them. Select the one that (a) is willing to take you on, and (b) you feel is most consistent with your answers to the questions above.

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