Dr. Kenneth Ribet of UC Berkeley visited campus to deliver our annual Kieval Lectures.  Professor Ribet is known for proving a key result that Andrew Wiles used to prove Fermat’s Last Theorem. I enjoyed his lectures and gleaned a little bit about elliptic curves and the modern process by which mathematical knowledge advances.  Just like after an argument has ended, one knows what one should have said, it was only when the lectures were over, that I knew what I wanted to ask Dr. Ribet.  Here are my post-hoc questions.

1.  Apparently what makes elliptic curves special is that points on the curves have a group structure the operation being drawing a chord through two points to get a third.   What is that about and why would anyone have thought of it?

2.  Wiles’ proof of Fermat’s Last Theorem was around 300 pages of dense mathematical reasoning.  Presumably with the benefit of hindsight, parts of the proof have been shorted and clarified.  The fact of Fermat’s Last Theorem seems to be some sort of collateral damage that resulted from the new theorems that Ribet and Wiles discovered.  Ribet’s result, if I remember correctly, was even a proof by contradiction.  Is there anything in the array of ideas used in the proof that clarifies our thinking about $a^n + b^n = c^n$?  Are there any new models the Dr. Ribet can provide us?

I think the second question is merely a plea by me for some way to think about $a^n + b^n = c^n$ that would not require, would that I were so capable, me spending 10 years of my life learning the necessary mathematics.

Anyway those are the questions that I wished I had asked Dr. Ribet.