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?