A geometry masterpiece: Yale prof solves part of math’s ‘Rosetta Stone’news.yale.edu/2024/11/01/geometry-masterpiece-yale-prof-solves-part-maths-rosetta-stone
430 sats \ 10 comments \ @south_korea_ln 3 Nov science
Sam Raskin has wrapped his head around a math problem so complex it took five academic studies — and more than 900 pages — to solve.
The results are a sweeping, game-changing math proof that was decades in the making. Working with Dennis Gaitsgory of the Max Planck Institute and a team of seven other mathematicians, Raskin has solved a segment of the Langlands Conjectures, long considered a “Rosetta Stone” of mathematics.
The Langlands Conjectures, named after Canadian mathematician (and former Yale professor) Robert Langlands, suggested in the 1960s that deep, unproven connections exist between number theory, harmonic analysis, and geometry — three areas of math long considered distinctly separate. Proving these connections, mathematicians say, could suggest ways to translate certain areas of math that had seemed dissimilar.
If i remember well, bridging these different fields was also what led to the proof of Fermat's last theorem by Andrew Wiles.
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136 sats \ 2 replies \ @SimpleStacker 3 Nov
Man, it seems to be one of those problems that can't easily be explained to non-mathematicians. I couldn't really get an understanding of what the Geometric Langlands Conjecture really was from reading the article.
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0 sats \ 1 reply \ @south_korea_ln OP 3 Nov
Reading this book aimed at the layman gave me some understanding as to what exactly Wiles achieved by bridging different fields of mathematics to prove Fermats theorem. But it was so long ago, that all I remember is the general idea and no specifics. I am by no means a mathematician.
The general idea is that by showing mathematical equivalence between two fields that are seemingly completely distinct, something that is impossible to prove in one field but easy to solve in the other field effectively gets proven through this equivalence. The Langlands program's mission is to prove these kinds of equivalences between fields. Hence the use of the Rosetta stone in the article i originally linked.
EDIT:
ChatGPT's input:
The Langlands program is a far-reaching set of conjectures and theories that seeks to establish deep connections between number theory and geometry through the framework of representation theory. One of its main goals is to demonstrate relationships between Galois representations (which arise in number theory) and automorphic forms (which are related to harmonic analysis and representation theory).
Andrew Wiles's proof of Fermat's Last Theorem is a remarkable example of this interplay. Wiles showed that proving Fermat's Last Theorem could be reduced to demonstrating a particular case of the Taniyama-Shimura-Weil conjecture, which is part of the Langlands program. This conjecture posits that every elliptic curve is related to a modular form. By proving this relationship, Wiles was able to prove Fermat's Last Theorem indirectly.
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0 sats \ 0 replies \ @SimpleStacker 3 Nov
That's inspiring, but way over my head! Math was always my favorite subject, I was just too chicken to pursue a career in math.
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21 sats \ 2 replies \ @didiplaywell 3 Nov
Fascinating! Thank you for sharing it!
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10 sats \ 1 reply \ @south_korea_ln OP 3 Nov
Math papers are of a different world. 900 pages of equations for just 5 publications. This wouldn't fly in my field ;)
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21 sats \ 0 replies \ @didiplaywell 3 Nov
While I admire this type of work, I find much more pleasure on simple and brief works. The simplicity in itself is enticing for me. I have a bunch of works I never published. I'm looking for a way to self host and own a blog domain to do so, I don't want to have my work dependent on a domain provider.
I'm unsure if the best way is just to use a Nostr account as a "domain" and a "blog", tough it seems the way to go.
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0 sats \ 2 replies \ @Rothbardian_fanatic 3 Nov
Was this what led to proving Fermat’s last theorem? Have those intersections and connections been explored and used, yet? It seems to me that the proof required a machine to complete it (or am I incorrect). Machine proofs were only fairly recently accepted as proving anything. In any event, It was a huge step forward.
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20 sats \ 1 reply \ @south_korea_ln OP 3 Nov
Yes, fermat is a special case of the Langlands program.
No, wiles' proof did not require computers other than what is fairly common for ultra complicated theoretical proofs: for verification purposes.
A famous proof that was achieved with computers is the map colouring one.
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0 sats \ 0 replies \ @Rothbardian_fanatic 3 Nov
As far as I know, the computer proofs run an enormous amount of iterations of the problem, perhaps regressively, until they reach a solution or contradiction. There was contention over whether the computer proofs were mathematically sound or not. I don’t have the expertise to weigh in on that controversy. I think it was settled that the computer proofs are acceptable to mathematicians.
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0 sats \ 0 replies \ @Cotton 3 Nov
Exciting stuff for anyone who follows big advances in math!
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