Inter-universal geometry と ABC予想 (応援スレ) 77

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 77 (1002ﾚｽ)
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614: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/11/09(日) 13:56:22.21 ID:QrKJGO9s

>>576
>IUTについては
>かみ合う議論が難しい

巡回ご苦労さまです
全くです
下記の 2020 by woit での Peter Scholzeと Taylor Dupuyとの論争も
結局は、かみ合わなかった

(参考)
https://www.math.columbia.edu/~woit/wordpress/?p=11709
Latest on abc Posted on April 3, 2020 by woit
Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by Publications of the Research Institute for Mathematical Sciences (RIMS) of Mochizuki’s purported proof of the abc conjecture.
This is very odd. As the Nature subheadline explains, “some experts say author Shinichi Mochizuki failed to fix fatal flaw”.

Peter Scholze says:
April 6, 2020 at 9:28 am
I have been weighing back and forth commenting again on this matter. However, the news in that last comment by David J. Littleboy convinced me that it might be good, even if futile, to say something again.

I may have not expressed this clearly enough in my manuscript with Stix, but there is just no way that anything like what Mochizuki does can work. (I would not make this claim as strong as I am making it if I had not discussed this for with Mochizuki in Kyoto for a whole week; the following point is extremely basic, and Mochizuki could not convince me that one dot of it is misguided, during that whole week.) It strikes deep into my heart to think that in the name of pure mathematics, an institute could be founded for research on such questions, and I sincerely hope that this will not come back to haunt pure mathematics.

The reason it cannot work is a theorem of Mochizuki himself.
略
I’m really frustrated with the current situation. What EricB reports from the Asahi Shinbun also sounds deeply troubling, effectively arguing along national lines; again, this strikes deep into my heart. I’m really quite surprised by the strong backing that Mochizuki gets from the many eminent people (who I highly respect) at RIMS.
If I can in any way help to mitigate the situation, I’d be most happy to.

Taylor Dupuy says:
April 6, 2020 at 1:00 pm
Hi Peter!
First, hope your pandemic is going well. Mine is going ok. Hard to get things done without daycare.
Second, let me say that for hyperbolic curves over a p-adic field K (with no extra hypotheses like strictly Belyi type or Belyi type or canonical lift) that pi_1(Z) “determines” Z is open. Also, I personally would advocate against using words like “determines”, “reconstructs”, etc that have been causing sooo many problems in discussing all of these things.

http://rio2016.5ch.net/test/read.cgi/math/1761878205/614

618: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/11/09(日) 14:24:55.46 ID:QrKJGO9s

>>614 補足
>下記の 2020 by woit での Peter Scholzeと Taylor Dupuyとの論争も
>結局は、かみ合わなかった

1）Scholzeちゃんも、若い
　「”radical simplifications”という明白な ストローマン論法」>>565
　が、彼は 若いから 数学外道だと 分っていない
　正しいと 信じ込んでいる。だから、Taylor Dupuyとの論争が かみ合わない
2）例えば O–Takegoshi L2 extension theorem https://en.wikipedia.org/wiki/Ohsawa%E2%80%93Takegoshi_L2_extension_theorem
　”・・・  Kensho Takegoshi in 1987,[1] using what have been described as ad hoc methods involving twisted Laplace–Beltrami operators, but simpler proofs have since been discovered.[2] ( Siu (2011))
　Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.”
3）このように、”simpler proofs have since been discovered”は、良いのです
　だが、「証明をsimplificationしたら 証明にギャップが出来ました」というN大の学生が居たら？
　O先生も「あ然、ぼう然、がく然」ｗ でしょう。O先生「おまえ 本当にN大 数学科生か？」とｗ
　つまり、”simpler proofs”は是なれど、”simplification→ギャップ出来た”は 原理的に不可なのです

この簡単な理屈が、若いScholze氏は 理解できていない！
彼は「証明をsimplificationしたら 証明にギャップが出来ました」という 非数学のディベート論法を
公式の数学主張として 持ち込んで 使った 史上初の数学者かも知れない

http://rio2016.5ch.net/test/read.cgi/math/1761878205/618

624: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/11/09(日) 16:02:17.60 ID:QrKJGO9s

>>537より 追加
https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence
Geometric Langlands correspondence
Status
A claimed proof of the categorical unramified geometric Langlands conjecture was announced on May 6, 2024 by a team of mathematicians including Dennis Gaitsgory.[7][8]
The claimed proof is contained in more than 1,000 pages across five papers and has been called "so complex that almost no one can explain it".
Even conveying the significance of the result to other mathematicians was described as "very hard, almost impossible" by Drinfeld.[9]

Breakthrough Prize なんだね（下記）

https://en.wikipedia.org/wiki/Dennis_Gaitsgory
Dennis Gaitsgory (born 17 November 1973) is an Israeli-American mathematician. He is a mathematician at Max Planck Institute for Mathematics (MPIM) at Bonn and is known for his research on the geometric Langlands program.
Honors and awards
In 2025, he received the Breakthrough Prize in Mathematics.[3]

（Scholzeさん）
https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/
Quanta magazine
Monumental Proof Settles Geometric Langlands Conjecture
Erica Klarreich July 19, 2024
The proof represents the culmination of three decades of effort, said Peter Scholze(opens a new tab), a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.”
Dennis Gaitsgory (left) and Sam Raskin led the nine-person team that proved the geometric Langlands conjecture.

https://breakthroughprize.org/News/91
Breakthrough Prize 2025 April 5, 2025 – Los Angeles
Mathematics
Dennis Gaitsgory wins the Breakthrough Prize in Mathematics for his central role in the proof of the geometric Langlands conjecture. The Langlands program is a broad research program spanning several fields of mathematics. It grew out of a series of conjectures proposing precise connections between seemingly disparate mathematical concepts. Such connections are powerful tools; for example, the proof of Fermat’s Last Theorem reduces to a particular instance of the Langlands conjecture. These Langlands program equivalences can be thought of as generalizations of the Fourier transform, a tool that relates waves to frequency spectrums and has widespread uses from seismology to sound engineering.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1761878205/624

625: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/11/09(日) 16:02:39.89 ID:QrKJGO9s

つづき

In the case of the geometric Langlands conjecture, the proposed one-to-one correspondence is between two very different sets of objects, analogous to these spectrums and waves: on the spectrum side are abstract algebraic objects called representations of the fundamental group, which capture information about the kinds of loop that can wrap around certain complex surfaces; on the “wave” side are sheaves, which, loosely speaking, are rules assigning vector spaces to points on a surface. Gaitsgory has dedicated much of the last 30 years to the geometric Langlands conjecture. In 2013 he wrote an outline of the steps required for a proof, and after more than a decade of intensive research in 2024 he and his colleagues published the full proof, comprising over 800 pages spread over 5 papers. This is a monumental advance, expected to have deep implications in other areas of mathematics too, including number theory, algebraic geometry and mathematical physics.

2025 New Horizons in Mathematics Prize
Sam Raskin, Yale University
For contributions to the geometric Langlands program, including the theory of the Whittaker model and the proof of the geometric Langlands conjecture in characteristic 0.

https://people.mpim-bonn.mpg.de/gaitsgde/GLC/
Proof of the geometric Langlands conjecture
This page contains five papers, the combined content of which constitutes the proof of the (categorical, unramified) geometric Langlands conjecture.
This is a collaborative project of D. Arinkin, D. Beraldo, J. Campbell, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin and N. Rozenblyum.
Papers:
GLC I: Construction of the functor
GLC II: Kac-Moody localization and the FLE
GLC III: Compatibility with parabolic induction
GLC IV: Ambidexterity
GLC V: The multiplicity one theorem
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1761878205/625

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