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?