How can I know if I’m working through my PhD correctly?

Question

I’m a math PhD student in a university in Europe, working in a highly abstract field. I’m halfway through the intended three years. But I have the feeling I’m not doing it right.

When I look back, two years ago I knew almost anything of the theory I’m working in. So from that point of view, it’s reassuring. But I am still learning. My supervisor has given me a problem which is not very definite or well-defined: I have to abstract some structure from some old, classical scenario and try to update it to some more modern objects, in hopes of gaining insight and computability.

This has proven a bit formidable to me, even if he said it shouldn't "take me more than some weeks". After trying the “hands-on” approach my supervisor had some months ago, I got to realize that it wouldn't work, at least not naively. So I started thinking of an easier case, and then the problem started to divert and I feel like I’m on very loose ground now.

The situation is the following: my supervisor has a very computational approach to math, whereas I err on the conceptual, abstract side. So when I started getting farther away from his line of thought, he progressively stopped being helpful to me, to the point where in our meetings I usually tell him what I have thought of or learnt, he nods, sometimes gives me an idea, but most of the times he just doesn’t help me. And the main problem is, I think, he doesn’t think about my problems at all outside of our short meetings (even when our meetings end up with a very well-defined, specific question from me). So it’s all up to me and I feel inept and insecure, especially because I try to apply stuff I learn on my own, stuff my supervisor doesn't really know, many times.

I also think that I’m thinking -too- conceptually, I think much more by analogy than by computation, but since I’m a beginner I don’t know if I’m really thinking about the objects like I should, but my supervisor doesn’t speak to me much so I’m confronting articles, books, monographs by myself. I have learned much more by myself than from him. It's been a long while since he's tried to transfer some knowledge to me (as I said, our meetings consist mostly of me showing my current problems). Honestly, I feel like I'm doing my PhD unsupervised, and I'm scared to be leading my mathematical thoughts into unfertile regions.

My question is: how can I know that I’m working through my PhD correctly? I feel aimless, and moreover for the moment I feel like I’m just juggling with other people’s knowledge. I keep incorporating knowledge that I try to juggle. To put it differently, I feel like I’m collecting by myself pieces of a puzzle, and I’m just putting them together, without creating any pieces. I’m just picking up other people’s fruits. I feel like whatever result I might end up converging to will be just a trivial consequence of some stuff done by others. When I read articles, I many times skip the proofs, and I feel like I know many things at a “moral” level. I really don’t have the time to go through the technicalities, and at the same time, they look so daunting! I always skip the hardest parts, stacking them into “black boxes”.

I have read other posts about the “impostor’s syndrome”. I certainly suffer from this and other related issues, but this question is not about coping with the issue “how not to feel worthless”, but rather “how can I be sure that I’m doing this right?”. I feel like this will just go on and it just doesn’t feel like anything will come out of my struggling. Am I wrong?

I am a hard worker and I have had good results getting here (as in, grades, scholarships, etc.). I have managed to learn a lot of different math successfully. I have a very good memory and an uncanny ability to find references, which is what my impostor’s syndrome tells me it’s what has been saving me; I feel like I am a good bibliothecologist, a good organizer, a good cleaner, but that doesn’t imply I’m a good creator; I have given no signs of being able to -create- anything. How should I cope with this?

Answer 1

My question is: how can I be know that I’m working through my PhD correctly?

Well, you can't as you haven't done it before, but in some sense you can never truly know anything, if that makes you feel better. In any case, there's no "right way" to do a PhD. Research is by nature a process where you wander into unknown territory, and at first you have no maps or guideposts. Then maybe you find what you are looking for, and maybe you don't, but as some measure of reassurance:

• After wandering around long enough, you can usually discover something new. Many times it may just be a different point of view or minor results, but at least you have understood something new. Still, usually when you spend a long time on a project you can get something publishable out of it in my experience.

Here are some other points regarding other things in your post:

• Many PhD advisors are very hands-off. It's also natural that they become less helpful as you go beyond the bounds of their expertise. However, it's probably not great if your advisor is thinks what you're doing is uninteresting. (I don't know if this is the case or not.) It's probably worthwhile to have a meta-research talk with him, including whether he thinks it's worthwhile to continue along these lines or consider a different approach.
• If you go out of the realm of your advisor's expertise, it might make sense to try talking to an expert who can give you better feedback. Conferences are one good opportunity for this.
• No professional mathematician has time to understand all or even most papers of interest to them. You need to learn to prioritize, which it sounds like you're already doing.
• You actually don't have to be that creative to do new things, or have too many new ideas, or be successful in research. Many PhD theses boil down to learning a bunch of stuff and then doing computations. This does not mean they are bad--many of them are very good. It's also hard to be creative before you reach a certain level of understanding, so you may not realizes your creativity till later, at which point you will almost invariably think, "How did it take me so long to come up with that? That's so simple."
• Organizational skills are not so separate from creativity. Many times problems can be understood easily after organizing your thoughts about them in the right way. This was part of the Grothendieck philosophy, and I doubt many mathematicians criticized Grothendieck for not being creative enough.
• While research can be frustrating and disheartening, you should generally enjoy learning and thinking about math, otherwise a PhD or academic track is not for you. Most people have good times and hard times with research. This is natural. I view one of my most important duties as an advisor is to encourage my students to help them get through the hard times.

Answer 2

This we can't really answer, without a truckload of information on your particular situation. Which we wouldn't read if we had it, anyway.

You should talk this through with your advisor, trusted faculty, fellow graduate students. Preferably all of the above.

Answer 3

My simple answer to your complex question is try to publish papers before hand. The reviews give you a good understanding of what is required, and also the publications can make your viva effectively a formality.