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I started a PhD program this September and I keep having doubts about whether or not I can achieve what I want being in this program and working with my current supervisors. I'm in the differential geometry group and I'm still learning semi-simple things, and haven't started a project yet. But from what I've seen so far, I think I'm more of a "visual" person if that makes sense- By that I mean I enjoy thinking about shapes and how they change, and if something like that is involved I'm motivated to do calculations. But without the motivation, calculations seem mostly dry and pointless to me. I always knew the first part (about enjoying thinking visually) but not the second part (that without this I don't feel motivated to do calculations.)

I know that it's possible to work in differential geometry and still enjoy some "visual" thinking, but the problem is that my supervisors' research involves mostly calculations. My fear is that I'll have to do more or less what they do, and not be able to deviate toward the stuff I enjoy more. So I guess one of my questions is: is that really the case? I worry that it is because at some point they will suggest a specific topic/question for me to work on which will turn into my PhD thesis and later career. (I know that in US PhD students have a year or two to figure stuff out before finding a supervisor, but I'm in Canada and here you pretty much have to chose what area you'll be working in before applying to the program.)

And if it is possible to deviate, is it actually possible to "make it" in a specific area if you don't have an expert in that area to guide you? By "make it" I don't mean winning a Fields medal or anything, just graduating from a PhD program and being able to do future related research.

I know that I should definitely talk to my supervisors about this but I'm afraid they'd say "leave the program then" because they kind of warned me about this (but I was too clueless to know what they mean back then.)

I would appreciate any advice on what I can/should do, especially if you've been in a similar situation.

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    Nearing 40 years since my undergraduate degree and I’m still finding cool new things to be interested in and contribute to using knowledge and techniques picked up over the decades. Frankly, you don’t know yet what you really like, but seem to have one good tool you like. You want a toolbox full of tools to have lots of fun!
    – Jon Custer
    Feb 17 at 16:18
  • @JonCuster Thank you for your comment! I think my current understanding of academia, at least in math, is that there are all these "clusters" of mathematician doing different types of math, and once you plant your roots in one, it's very hard/nearly impossible to leave and start over in a new cluster. So I'm worried that if I start in a cluster that's not right for me, I would get stuck there. I understand your point about the tools and agree that I don't know what I really like yet, but will I ever figure it out being in the wrong cluster?
    – Gordafarid
    Feb 18 at 10:04

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Don't let short term thinking dominate your long term interests. A doctoral program should actually teach you things you don't know rather than just confirm the things you already know (and love). Yes, you may have to put your geometric thinking a bit toward the back of your mind for now, but it would be a mistake to give it up. It will probably provide insight to the more computational things you are finding in the work of your advisors.

My field was analysis (classical real) and the insight I gained from a deep and visual understanding of real functions helped with the details. "Wow, a function that is continuous everywhere but nowhere differentiable. Wow." "Wow, a function that takes every value in every neighborhood of zero. Wow."

Yes talk to your supervisors. Yes, keep your geometric intuition. Get the doctorate behind you and get yourself into a tenure-able position. Now you can refocus on the geometric if you like. It might even be an advantage in your teaching if you can communicate that insight.

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  • I didn't know there is a function that takes every value in every neighborhood of zero...would you mind providing a reference? (one not too complicated if possible) Feb 17 at 14:27
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    (1/x)sin(1/x) is all you need @DavidRaveh, Wow. It actually takes every value infinitely often in every neighborhood of zero. Double Wow.
    – Buffy
    Feb 17 at 14:33
  • Wowwwwwwwwwwwww Feb 17 at 14:40
  • Yeah. It sort of slams into the y axis. It also gives insight into non-analytic complex valued functions of a complex variable. Something I find hard to "visualize" there.
    – Buffy
    Feb 17 at 14:41
  • @DavidRaveh no, Wowwwwwwwwwwwww Wowwwwwwwwwwwww!
    – Jon Custer
    Feb 17 at 16:06

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