Has anyone studied Mathematics this way? What were your results?

Question

If you open the average text of Math, the reason for the importance of the ideas in textbook seem to be lost. For eg, the weil conjectures motivated a lot of the algebraic geometry ( there is a nice wiki entry on this).

If we think of the Mathematical learning process for a given topic as a hero's journey, then I think having a big problem that one can comprehend, but can not solve with their current tools, which one can work toward for, can be really motivating.


Does anyone learn this way? What were your results?

Answer 1

Back in undergrad I was not a very good student but I think a good part of why I kept trying was because I was fascinated by Abel-Ruffini, it was crazy to me that such a theorem could exist and my brain couldnt understand how do you prove that there is no formula, how would they know!? So I had to learn a lot of algebra to understand this.

I took a second course in abtract algebra and we proved the theorem and honestly it was so underwhelming, didnt feel like it was worth it. But I guess I was already on my way so I did end up finishing the degree.

Answer 2

I study math this way, and many people I know do--I think most graduate students shift to studying math this way since once you have a basic foundation built up, problem driven learning is better (since now you don't need to know what's on the test--you need to know enough to succeed in research, and so you need to make sure you learn valuable tools that come up often).

You mention the Weil conjectures; they might be ambitious for a first course in algebraic geometry. As a first attempt to learn, say, the material in Hartshorne, I think it would serve one well to try and understand the following problems.

1. Let k be a field. How do you classify nonsingular projective curves over k? This is a great question, since it forces one to grapple with how to define 'nonsingular curve' over a general field. I imagine one would first want to construct all \*proper\* curves over k, and then learn about linear systems to see how to construct a projective embedding of a proper curve.
2. Prove the Weil conjectures for algebraic curves--Weil did this using clever but classical arguments.