6 ms·

Ever since my undergraduate I've wanted to understand what representation theory is and the rough outline of how Andrew Wile's proof was constructed. This arti

Ever since my undergraduate I've wanted to understand what representation theory is and the rough outline of how Andrew Wile's proof was constructed. This article gave me both in a very understandable and engaging way. Thank you to the author!!!

quickthrower2 6y ago
I still want to know what was the proof Fermat wrote was that couldn't fit in a margin.
- dllthomas 6y ago
  It seems there's a fair chance it was a flawed proof, in which case I doubt we'll ever be particularly confident although we could possibly surface some candidates.
  - DC-3 6y ago
    Something I always wondered is if we know of the existence of a proof that is both simple and also non-obviously flawed and as such could have been Fermat's solution?
    - pathsjs 6y ago
      Yes, Lamé solution, which assumes unique factorization in the rings of integers of cyclotomic fields, and can in fact be salvaged to prove the case of regular primes
      - quickthrower2 6y ago
        Thanks while I don't fully understand what this means I think I get the idea. I assume it's a proof that has the "n" over infinite many cases (primes) but not all of them? Googling it was hard because there are other Lamé things. I'll just imagine he wrote this proof. Not the full proof, but not lame!
        pathsjs 6y ago
        Yes, for some details look at http://fermatslasttheorem.blogspot.com/2006/01/lams-proposed-proof.html http://fermatslasttheorem.blogspot.com/2006/01/lams-proposed... or https://math.stackexchange.com/questions/953462/what-was-lames-proof https://math.stackexchange.com/questions/953462/what-was-lam...
- mellosouls 6y ago
  Despite Fermat's brilliance, it's barely feasible that such a proof exists after 300 years of attempts by professionals and amateurs alike. Ironically though, the tantalising idea of the existence of a simple proof - motivated by that infamous quote - contributed massively (along with the simplicity of the conjecture itself) to the theorem's notoriety and the myriad attempts to solve it.
  - ubercow13 6y ago
    Fermat was a brilliant social engineer as well as mathematician.
    - rrmm 6y ago
      It's the predecessor of the dictum about putting a wrong answer on the internet to get the right answer. Sorta, and with a few hundred years involved.
    - QuesnayJr 6y ago
      He wrote the note to himself, a long time before he died. He then publicly announced a proof for the n=4 case, and nothing about the general case. It strongly suggests that he realized his proof was wrong.
sdenton4 6y ago
A somehow-much-less-popular way to come at representation theory is via generalized Fourier transforms. The irreducible representations are basically different "frequencies" which you can use to decompose functions on a group. And in fact, all of the major Fourier theorems carry over perfectly... There's even analogues of the fast Fourier transform, via chains of subgroups.
- memexy 6y ago
  Are there any good references for this approach (via generalized Fourier transforms)?
  - kcoul 6y ago
    I wouldn’t call this 3blue1brown video a reference, but until I watched this (coming from an audio background), I didn’t quite grasp that a continuous stencil of any complex silhouette could be described in terms of a set of contributing vectors. Helped click that there is a perfect visual analogy to the deconstruction of complex audio to sine waves. https://youtu.be/r6sGWTCMz2k https://youtu.be/r6sGWTCMz2k
  - pathsjs 6y ago
    Yes: Folland, A course in abstract harmonic analysis, CRC press
  - bollu 6y ago
    There is a thesis by Kondor: "Group theoretical methods for machine learning" - https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf which provides links to the associated literature. The documentation of his library (SnOB) is great: http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf introductory slides on representation theory in machine learning: http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.pdf http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.p... Full mini course: http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html
dhosek 6y ago
There's a good undergraduate-level text that gives enough background to understand the proof. http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek

quickthrower2 6y ago

I still want to know what was the proof Fermat wrote was that couldn't fit in a margin.

dllthomas 6y ago
It seems there's a fair chance it was a flawed proof, in which case I doubt we'll ever be particularly confident although we could possibly surface some candidates.
- DC-3 6y ago
  Something I always wondered is if we know of the existence of a proof that is both simple and also non-obviously flawed and as such could have been Fermat's solution?
  - pathsjs 6y ago
    Yes, Lamé solution, which assumes unique factorization in the rings of integers of cyclotomic fields, and can in fact be salvaged to prove the case of regular primes
    - quickthrower2 6y ago
      Thanks while I don't fully understand what this means I think I get the idea. I assume it's a proof that has the "n" over infinite many cases (primes) but not all of them? Googling it was hard because there are other Lamé things. I'll just imagine he wrote this proof. Not the full proof, but not lame!
      - pathsjs 6y ago
        Yes, for some details look at http://fermatslasttheorem.blogspot.com/2006/01/lams-proposed-proof.html http://fermatslasttheorem.blogspot.com/2006/01/lams-proposed... or https://math.stackexchange.com/questions/953462/what-was-lames-proof https://math.stackexchange.com/questions/953462/what-was-lam...
mellosouls 6y ago
Despite Fermat's brilliance, it's barely feasible that such a proof exists after 300 years of attempts by professionals and amateurs alike. Ironically though, the tantalising idea of the existence of a simple proof - motivated by that infamous quote - contributed massively (along with the simplicity of the conjecture itself) to the theorem's notoriety and the myriad attempts to solve it.
- ubercow13 6y ago
  Fermat was a brilliant social engineer as well as mathematician.
  - rrmm 6y ago
    It's the predecessor of the dictum about putting a wrong answer on the internet to get the right answer. Sorta, and with a few hundred years involved.
  - QuesnayJr 6y ago
    He wrote the note to himself, a long time before he died. He then publicly announced a proof for the n=4 case, and nothing about the general case. It strongly suggests that he realized his proof was wrong.

sdenton4 6y ago

A somehow-much-less-popular way to come at representation theory is via generalized Fourier transforms. The irreducible representations are basically different "frequencies" which you can use to decompose functions on a group. And in fact, all of the major Fourier theorems carry over perfectly... There's even analogues of the fast Fourier transform, via chains of subgroups.

memexy 6y ago
Are there any good references for this approach (via generalized Fourier transforms)?
- kcoul 6y ago
  I wouldn’t call this 3blue1brown video a reference, but until I watched this (coming from an audio background), I didn’t quite grasp that a continuous stencil of any complex silhouette could be described in terms of a set of contributing vectors. Helped click that there is a perfect visual analogy to the deconstruction of complex audio to sine waves. https://youtu.be/r6sGWTCMz2k https://youtu.be/r6sGWTCMz2k
- pathsjs 6y ago
  Yes: Folland, A course in abstract harmonic analysis, CRC press
- bollu 6y ago
  There is a thesis by Kondor: "Group theoretical methods for machine learning" - https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf which provides links to the associated literature. The documentation of his library (SnOB) is great: http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf http://www.gatsby.ucl.ac.uk/~risi/SnOB/SnOB/SnOB.pdf introductory slides on representation theory in machine learning: http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.pdf http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/symmetric.p... Full mini course: http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html http://www.gatsby.ucl.ac.uk/~risi/courses/mini08/mini08.html

dhosek 6y ago

There's a good undergraduate-level text that gives enough background to understand the proof. http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek