Do not have the right tools for my thesis problem

Question

I am pursuing a PhD in analysis and partial differential equations. My advisor has given me a problem which actually turns out to have a very large algebraic geometry requirement. Although my analysis is very good, my geometry and algebra have always been a little weak. I find the problem very fascinating and, surprisingly, the problem can even be stated in a way that even a first year undergraduate could understand it.

Now, I'm essentially stuck. I do not know if learning all the geometry that I never learned before would help me out. In fact, I doubt it. The other thing is that I feel that for a problem that can be stated in simple linear algebra / calculus terms, such weighty tools might not even be required and some kind of partial results should be obtainable with whatever knowledge I have.

The fact that I am not making any real progress is making me feel very insecure and making me question whether I can become a mathematician. I need some advice.

Should I set myself a time frame and bundle up things as well as they can be bundled up?

Should I set myself the challenge of learning all the geometry that may/may not work?

Should I quit?

Answer 1

As with everything else about being a graduate student, the correct answer is always: Talk to your adviser. She will likely have a much better overview about the field, the techniques needed, what you probably will need to make progress with your research, and many other things. Discuss with her your thoughts about your uncertainty where the right direction lies, what this means for your career, whether you're qualified, etc. Communicate!

Answer 2

I had exact same problem in a different domain. I think this little story of mine will help you:

6 Months Before Thesis Submission: I was almost done with the whole thesis, then suddenly the supervisor decided to force me into adding another aspect of my work to my thesis. Basically another chapter on the whole matter that I already solved and concluded, but with a different tone (as it was another domain parallel to the domain that I solved).

Panicking Period (1 month): I panicked like hell. The whole thing was done and she wanted me to go to a domain that I ignored since I put my foot at a university.

Exit Strategy (1 month): The time was running out, I said to myself, well I will find 'easy references' that tackled this issue in a very simple way, will learn them, then write the chapter about it and then I will cite them. I found around 20 references on the subject matter.

Gold mining (1 month): It was one the most fun period of my PhD!. I narrowed these 20 references into 1 (thats right!), and then learned its related references.

Writing Period (2 months): Wrote around a 35 pages based on that reference and my work. It actually gave depth to my thesis, as I explained and tackled the same problem using a different method.

Putting my foot down and submission (1 month): Let the game of cat and mouse begin!. Supervisor did not liked the approach I was taking. I did explain that it is actually a very good and important reference, and this chapter is enough on the matter. She disagreed and I did not gave up.

Defence: Fortunately or unfortunately, one of my examiners where a master in the field I tackled and wrote a chapter about. He loved it. In fact 70% of the defence time was about the material I wrote on that chapter.

Conclusion: Be honest with yourself. At the end of the day, if you can defend what you wrote and contributed, then well, you be fine in defending your thesis and your whole PhD.

Answer 3

My experience in the matter (computer science): never be afraid to learn.

Every new concept opens a lot of new possibilities. Try to see the problem from different points of view, you will often find that there are similar problems and this change will almost always help you develop a solution.

Of course, it wouldn't be wise to charge full time into each possibility... hedge your bets... you might want to focus on the pragmatic approach first, separating a few hours a week to study new stuff.

Just to join the chorus "talk to your advisor". They usually know what is best for you.

Answer 4

It wouldn't be a bad thing for you, if we take the long view, for you to learn at least some of the "geometry you never learned before". So even if the geometrical approach doesn't yield anything useful for your current research problem, you'll still have become a better-rounded mathematician.

Research often involves building a rope ladder into the nothingness, not knowing how long the ladder needs to be to reach the other side, not knowing whether your type of ladder will even be able to hook onto the other side once you get there.

Try to relax and enjoy this period of your life, in which you have the freedom to spend some months learning about something, trying something out, out of intellectual curiosity. Will it prove helpful? Will it prove to be a waste of time (in terms of your original research problem)? Will it turn up some little, or big, surprises? Will it turn out to be helpful for something else you're doing 10 years from now? Who knows. Enjoy the fact that no one is breathing down your neck saying, on a weekly basis, Did you solve it? Did you solve it?