Andrew Wiles: Fermat's Last theorem: abelian and non-abelian approaches

The successful approach to solving Fermat's problem reflects a move in number theory from abelian to non-abelian arithmetic.

This lecture was held by Abel Laurate Sir Andrew Wiles at The University of Oslo, May 25, 2016 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.

Program for the Abel Lecture 2016
1. "Fermat's Last Theorem: abelian and non-abelian approaches" by Abel Laureate Sir Andrew Wiles, University of Oxford
2. "Andrew Wiles' marvelous proof" by professor Henri Darmon, McGill University
3. "What is the Birch--Swinnerton-Dyer Conjecture, and what is known about it?" by professor Manjul Bhargava, Princeton University
4. "From Fermat's Last Theorem to Homer's Last Theorem" - a popular lecture by Simon Singh, author of Fermat's Last Theorem among other achievements. This lecture will never be published because the presentation contained material protected by intellectual property.

[APJS]

He is a very good lecturer as well as a clearly brilliant mathematician.

[XMIRC]

The mere pursuit of this problem has advanced mathematics so much - Thank you Fermat.

[AlonsoRules]

finally a video of Wiles talking about the proof

I only understand the initial problem lol

[xyzct]

If this doesn't get the chicks, I don't know what will.

[douglasstrother]

"We're gonna need a bigger margin.", Fermat.

[Wondering_human]

I'm in 9th grade so I didn't understand half of the stuff, but I promise I'll be back after I learn enough to understand this whole video. Prolly in 5-10 years but I'll brb!

[mertaliyigit]

Its extremely thrilling to watch this video after reading Simon Singh's book about it.

[valtinho-chefedesegurancad]

Andrew is incredible. I imagine how hard this was, even for Fermat. In any case, we can't say Fermat demonstrate that. He actually din't know how to do that!

[emmetray]

Did not understand anything after a^2 + b^2 = c^2, BUT very interesting :D

[helo]

I don't understand anything and when I finished the video I feel like I got better at math.

[fitofito]

I am surprised there are over 150k views so far. It was said that there were only about 10 people in this world who fully understood his proof when it was first published

[mitri]

"The proof is trivial and left as an exercise for the reader."
The proof:

[u.v.s.]

I understand everything except one little fine point.

Why do they give him ABEL prize for NON-Abelian approaches!????

[hwhefnbebe]

I’m dumb as rocks idk why I watched this 😂

[BuleriaChk]

In oreder for the multiplication operator to exist, both its elements must exist.
Russell's Paradox: 1^2 <> 1
# = 2 = 1+1 (first order)
Then #^2 = (1 + 1)^2 = [1^2 + 1^2] + [2(1)(1)] = 4(1^2) (second order - via Binomial Expansion)
where the first term is existence and the second is interaction (multiiplication, entanglement, entropy)
Note that existence and interaction are not 4D (1, 1, 1, 1) which diagonal is 4 elements without multiplication.

Every number is prime relative to its own base. n = n(n/n) = n(1_n)
Goldbach's Theorem: every even number is the sum of two primes: n + n = 2n
n is odd.
Godel's characterization of wff's in his meta-language only uses odd numbers (products of primes).
Therefore, the sums of odd numbers (even numbers) cannot be represented by hhis wff's.
So it is just Goedel's meta-language that is incomplete, not positive real numbers.

Together with Fermat's Last Theorem (applied to multinomials of arbitray powers), the arithmetic system is complete and consistent for positive real numbers.

There are no negative numbers:
-c = a - b, b > a
b - c = a, a + 0 = a, a - a = 0..

If there are no negative numbers, there are no square roots of negative numbers.

Proof of Fermat's Theorem for Village Idiots (n>1)
c = a + b
c^n = a^n + b^n +f(a, b, n) (Binomial Expansion)
c^n = a^n + b^n iff f(a, b, n) = 0
f(a, b, n)<>0
c^n <> a^n + b^n QED
Also valid for n > 1
c^2 = [a^2 + b^2] + [2ab]]
2ab < >0
c^2 <> a^2 + b^2 QED
(Pythagoras was wrong; use your imagination)

Check out my pdfs in physicsdiscussionforum "dot" org.

[trankt]

Speaking of prime, how expensive is Amazon prime?

[SanderBessels]

The story is definitely beautiful and interesting, but I think the general public could be informed better on the general theory that is behind the proof of FLT.

The relation between elliptic curves and modular forms is the real breakthrough.

Translating concepts from one field of mathematics into concepts from another field, preserving the structures, is something really powerful. It happens a lot, also in more familiar parts of mathematics (for instance a graph, a table and an equation).

You can even extend the analogy and talk about translating an old language using the Rosetta Stone of even discovering that the big religions of the world talk about similar things, but they use different words.

These discoveries are always amazing and I think people can appreciate it, no matter how well you understand the mathematics.

[SA]

Because of this video, proving FLT now looks simpler. Although it is very hard mathematics.

[daniboist]

The is the only guy that can say he was surprised Fermat was confused by something elementary. @7:10

[fans]

It would be great if he can talk about how he solved the modularity conjecture. Is there another lecture where he did that?