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I am about to become a Ph.D student in mathematics. I am having some doubts about where I should do my Ph.D and I would appreciate some advice, or anything really.

  1. I could do research with a very competent professor that is internationally well-known and is also very interested in me succeeding, having helped me with much already. He is also a really kind person. The area he is working in is one of the most fascinating areas in all of mathematics to me. However, it is in a different city from where I live now. As I live now, I have a really good apartment and a girlfriend too who lives here. She would be OK if I moved, although a bit sad. I am afraid of moving to a new city, and more than that, also somewhat scared that if I am not as good as my advisor belives me to be, it will all go terribly wrong.
  2. I could do research where I live. However, the advisors are not as competent as he is, nor as interested in me (so far, I am not their student yet). Further, the areas are not as fascinating (although very fascinating!). I could continue with living my life as of now.

So, what should I think about regarding these choices when it comes to making a future career? I am a bit vague here, since I want general advice to think about.

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    You've mentioned a number of considerations about what to look for when choosing a Ph.D program, but I'm not sure that any of us can help you make your decision. For some people, going to a prestigious institution is worth it and helps them. For others, that sacrifice of friends and family is too much for them to bear. You know yourself better than the rest of us! – Irwin Mar 19 '13 at 22:08
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    If your girlfriend is willing to move with you, then you've already conquered the biggest barrier to a more challenging and rewarding situation. – Dave Clarke Mar 20 '13 at 14:05
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So I'm just some random stranger on the internet, but you asked for advice, so here it is. Given only the information in your post, I think you have a difficult choice ahead of you:

Move or propose.

From a strictly professional standpoint, you make a very strong argument for moving. Having a more competent and more supportive advisor, and working in a field that excites you more, will have a tremendous impact on your several-decades-long mathematics career. As Dan C writes elsewhere, if you want to do math for a living, you should enroll in the strongest PhD program you can. (I'm assuming here that you actually want a career in mathematics; otherwise, getting a PhD in mathematics is kind of stupid.)

From a personal standpoint, you make a much weaker case for staying put. It's just another city; you'll be fine. You can always find another apartment. Do not listen to the Impostor Syndrome; everyone is afraid that they're not good enough. Your girlfriend is OK with your moving. There's some possibility that your current relationship won't survive the move, but the fact that you call her your girlfriend, and not your wife or even fiancée, suggests that you don't really think of the relationship as permanent anyway.

Yeah, I know, that's harsh. And maybe I'm totally off-base. Maybe your current relationship really is more important to you than the rest of your professional life. If that's true, tell her so.

Who knows? Maybe you can do both!

But whatever you decide, don't base your decision on fear. Instead, ask yourself: What do you really want? What really matters to you? Fear of change and Impostor Syndrome are completely natural, and far more common than most people think, but they're just voices in your head. Letting them control your decisions is not healthy. You have people in your life who believe in you, who support you, who want you to succeed and be happy, and who know you far better than Some Random Stranger on the Internet—listen to them!

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    +1 for not letting fear dominate your decision. And when it comes to academic topics, the people on academia are far from random, they are strangers to the OP though. – Paul Hiemstra Mar 20 '13 at 17:36
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    To be fair, the OP doesn't make clear whether the move is so far away that the choice does become "move or propose". After all, people do maintain long-distance relationships (especially if it's not that long) – Suresh Mar 20 '13 at 20:34
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    @Suresh Yep, my wife and I did that for five years. I do not recommend it. – JeffE Mar 21 '13 at 4:59
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    I did for three :) – Suresh Mar 21 '13 at 6:33
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You mention your "future career", and the answer to your question also depends on how you consider your career after your PhD. If you don't want to stay in Academia, and go for instance working in a bank or in some international organisation, then it's likely that the content of your PhD or the quality of your advisor does not really matter. In that case, if you think you can reasonably enjoy working where you currently live, it might not be worth changing location.

On the other hand, if you want to stay in Academia, then it's very likely that you will keep facing the problem you're facing now. Perhaps you'll be lucky, and the best PhD topic will be in the place you are, and then the best Postodc, and then the best Assistant Prof. position, and then the best Prof. position ... but it's quite likely that you might need to move, at least if you want to get the best position. I know few academics who work in the same place where they've done their graduate studies. I'm not saying it's impossible, but if you want the best opportunities, you might need to consider, at some point in your career, to move across the world.

There is no easy answer, and it's entirely up to you, but if you don't have a clear priority yet between going for what you might think is the best work opportunity for you and your personal/private life, you might not enjoy Academia ... (this is of course a pessimistic view, and plenty of academics have a remarkable career and a great personal life! but I'd rather point out the negative aspects so that you won't be disappointed later).

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It's hard for anyone to tell you what to do. Ultimately your decision comes from within your own blend of views, feelings, and priorities.

But there are common patterns of thought that we've all encountered, and it helps to understand how others have dealt with them.

Fear of change.

This is a very natural feeling to have. Whenever you're thinking of moving out of a comfort zone into a new situation, it's easy to focus on the fear of the unknown, because there's nothing else to focus on. In your case, there's fear of the move to a new city, and fear of how this will affect your relationships.

One way of addressing this is to try and find out more about the new situation and how the change will affect you.

  • where will you live ?
  • what will your work schedule be like ?
  • what will your office look like ?
  • how will you and your girlfriend manage the distance in the relationship ?

and so on. Right now, your ability to make a decision is clouded by uncertainty. Thinking concretely about these questions will help make the new scenario more real, and you'll be in a better position to evaluate whether the move is worth it or not.

Imposter syndrome

The other fear you've expressed is that "your advisor will discover that you're not as smart as you/they thought". This feeling is called the Imposter Syndrome and is typically associated with academics or those preparing for advanced study. Ironically, your feelings of inadequacy might qualify you already !

But the imposter syndrome is just that: it's a psychological filter that's caused by a focus on negatives and a dismissal of positives. It sounds like your advisor-to-be is very supportive and wants you to succeed. While it's possible that you might find a Ph.D is not for you, it's also entirely possible that you'll do very well for yourself. What will make the difference is not some fate stamped on your head the day you were born, but the work you do, the effort you put in, lots of luck, and many things that are beyond your control. Here again, getting more information can help - if you haven't already done this, maybe you can talk with the prospective advisor and map out what the first year or so of your Ph.D might look like.

Ultimately whether or not you choose to move depends on many factors, and there's no wrong answer here. But it's helpful to recognize where some of the thoughts you express might be coming from.

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I can't make a definitive recommendation one way or the other here. What I can tell you is that you need to consider the following:

  • Don't change jobs to get away from your present situation. Running and hiding somewhere else is neither productive nor psychologically healthy. (In other words, make sure you are going to another opportunity, not walking away from a situation—although it doesn't sound like that this is an issue here.)

  • Is the cost of what you're leaving behind worth what you're gaining? You seem to like living where you are. Is the opportunity for you to work with the big-name advisor important enough for you to give that up?

  • Before beginning a PhD, make sure that it's something you really want to do. Is the project you will be working on something that fascinates you enough that, even if your work doesn't go well for a long time, you'd still want to go in day after day to get things back on track?

  • Let me turn the first two points on their heads. Don't stay put out of fear. Is the cost of what you'd give up by staying worth what you'd gain by moving? – JeffE Mar 20 '13 at 15:41

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