4 ms·

I was obsessed with trying to understand it ever since I read the book. How many years of math studies one need to make sense of it? (I'm doing a CS master deg

I was obsessed with trying to understand it ever since I read the book. How many years of math studies one need to make sense of it? (I'm doing a CS master degree but I'm not even close to feeling that I know anything about math, imposter syndrome or more likely not enough math skills...) Any chances? Or should I probably give it up?

cokernel 11y ago
It took Wiles seven years to prove the thing (and the proof needed to be patched!), so I wouldn't be discouraged if it takes a while to understand. A nice thing about mathematics is, it waits for you. If you're interested in the mathematics behind this, I'm not sure a direct attack on the FLT proof is the best route to take. The paper that proves a famous conjecture is normally sitting on a mountain of prior work, which means the final paper (1) assumes familiarity with that mountain and (2) is highly technical because all of the understandable things have already been tried. So instead, why not start learning about the mountain? The truth of FLT follows from the two claims: (1) Taniyama--Shimura--Weil conjecture: "Every elliptic curve is modular." (2) Ribet's Theorem: "If FLT has a counterexample, then such and such an elliptic curve is not modular." As it happens, TSW was originally believed to be too difficult to prove, but I suppose the connection with FLT motivated people. Taylor and Wiles proved the absolute minimum of TSW that they could get by with and still get a contradiction from Ribet's theorem. (My understanding is that TSW is now fully proved -- the "modularity theorem".) If you're wanting to "get" FLT, I'd encourage you to look into elliptic curves, modular forms, and their relationship. I wonder if working on the mathematics of elliptic curve cryptography might be a good way to get a feel for elliptic curves. However, if you do want to take the direct route, I believe that Faltings's highly compressed article provides a syllabus. Once you can read and completely understand every sentence of that article, it is highly likely that the Wiles and Taylor and Wiles papers will make sense. I... would really not recommend this route.

cokernel 11y ago

It took Wiles seven years to prove the thing (and the proof needed to be patched!), so I wouldn't be discouraged if it takes a while to understand. A nice thing about mathematics is, it waits for you. If you're interested in the mathematics behind this, I'm not sure a direct attack on the FLT proof is the best route to take. The paper that proves a famous conjecture is normally sitting on a mountain of prior work, which means the final paper (1) assumes familiarity with that mountain and (2) is highly technical because all of the understandable things have already been tried. So instead, why not start learning about the mountain? The truth of FLT follows from the two claims: (1) Taniyama--Shimura--Weil conjecture: "Every elliptic curve is modular." (2) Ribet's Theorem: "If FLT has a counterexample, then such and such an elliptic curve is not modular." As it happens, TSW was originally believed to be too difficult to prove, but I suppose the connection with FLT motivated people. Taylor and Wiles proved the absolute minimum of TSW that they could get by with and still get a contradiction from Ribet's theorem. (My understanding is that TSW is now fully proved -- the "modularity theorem".) If you're wanting to "get" FLT, I'd encourage you to look into elliptic curves, modular forms, and their relationship. I wonder if working on the mathematics of elliptic curve cryptography might be a good way to get a feel for elliptic curves. However, if you do want to take the direct route, I believe that Faltings's highly compressed article provides a syllabus. Once you can read and completely understand every sentence of that article, it is highly likely that the Wiles and Taylor and Wiles papers will make sense. I... would really not recommend this route.