Monday, September 24, 2018

Atiyah's talk on the proof of Riemann Hypothesis scheduled for Monday

Hot, Monday 7:55: Before the 9:45 talk (watch video here and press "triangle play" in the lower left corner if paused; schedule; Laureates' server is overloaded), see this five-page paper by Atiyah. It starts with crackpot-style comments on the fine-structure constant but don't stop too early. The 15-line-long proof by contradiction using his "Todd function" (defined in another paper, thanks, pis.) is on Page 3. The strategy of halving the imaginary part of the Siegel root is exactly what I predicted on Quora! However, I don't understand his "Todd function" that is polynomial in convex sets of the complex plane but not in general. Can't it be proven that a regionally analytic polynomial function is polynomial everywhere? Oh, I see, it's just "weakly analytic". I must see what it means because he seems to mix real and complex analytic functions.
Originally posted on Fri Sep 21st morning

In 2012, Šiniči Močizuki claimed to have a proof of the \(abc\) conjecture. Now, exactly six years later, his proof – distributed over 500+ pages of text, not counting some "background" in additional 500+ pages of text – remains disputed. Some mathematicians claim that it has to be correct but they seem to be "insufficiently independent" of Močizuki. The truly independent ones remain silent or... negative.

In particular, the Quanta Magazine says that Jakob Stix and (the young, celebrated, fresh Fields Medal winner) Peter Scholze claim that they have isolated an unbridgeable gap in the Japanese proof. They met with Močizuki. The two sides couldn't agree. Scholze was just a "cheeky Hun who just barely jumped out of a vagina", Močizuki was a "brownie, gook, and nip", you can imagine that the exchanges between mathematicians keep their highest standards of diplomacy.

I think that this controversy is similar to some controversies in theoretical physics, perhaps including the "de Sitter space in string theory" controversy. In principle, it could just mathematics where everything is clear. But it's complex enough, with a potential for mistakes and some room for replacing detailed solutions by philosophies, so that people may end up believing in very different answers.

But frankly speaking, I don't really care about the \(abc\) conjecture – and I have never cared. So I wasn't ever tempted to start to read the 500 pages.

Óscar Santiago Gómez has pointed out a text by John Cook Consulting which talks about something that I care about a lot.

Next Monday, there should be a talk by Michael Atiyah, a top mathematician, who should talk about... his proof of the Riemann Hypothesis. What the ideas should be? He's claimed to have a
"radically new approach … based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928)."
The choice of the background makes it sound rather conservative – and I like it.

Two papers seem to be the Dirac equation (the year works fine – Atiyah-Singer theorem is about the index of Dirac-like operators), the Hirzebruch-Riemann-Roch theorem (the year works, and it makes sense – Riemann-Roch is about locally holomorphic complex functions with prescribed zeros and allowed poles). See a page by Atiyah where both papers play a role. Just to be sure, the Atiyah-Singer theorem is a generalization of the Riemann-Roch theorem, too, but he will probably use a more Hirzebruchy one.

Von Neumann (1936) may be regular rings (unless it's related to the Turing machines, the same year, which I find too far from the RH but I could be surprised; or some presentations of quantum mechanics by von Neumann in 1936 which seem too unoriginal). For years, we could have speculated whether the problems would be solved by a newbie or one of the most famous mathematicians. If this work existed and were correct, the second answer would be right.

Here you have a tweet by an organization serving the Abel Prize winners:

The word "presenenting" doesn't sound too persuasive – it's similar to "covfefe" – and the date also seems to be wrong but I wouldn't dismiss the rumor just because of these two typos.

Here you have the abstract that someone turned into a screenshot:

OK, I admit that I have thought that the third floor of the "new auditorium at the new university" is a prank. But I don't want you to miss the story in the case that it is not a prank LOL. The issue is that the New and Old University do exist in Heidelberg and a new auditorium was donated by an ex-ambassador to America.

I've spent hundreds of hours with efforts to prove the Riemann Hypothesis. Most of these efforts were basically variations – sometimes e.g. string field theory variations – of the Hilbert-Pólya program. In recent years, I changed my mind and became a Hilbert-Pólya skeptic. The program has proposed to prove the Riemann Hypothesis by showing that there is a Hermitian operator \(L\) whose eigenvalues are the imaginary parts of the Riemann zeta roots.

I still think that there may exist extremely natural operators \(L\) with this property but I no longer think that the existence and Hermiticity of \(L\) directly leads to the proof of the Riemann Hypothesis. Why? Because aside from the normalizable eigenstates, \(L\) may also have non-renormalizable "quasinormal modes" with eigenvalues away from the real axis. So the Riemann Hypothesis claim may reduce to the question whether all formal solutions are normalizable, and this question may be as hard as the original Riemann Hypothesis.

Lots of proofs have been wrong so even if the new auditorium exists and Atiyah presents a talk over there, it may be incorrect, too. It's probably more likely to be incorrect than correct. But it's Atiyah, so I won't dismiss it out of hand. Instead, I plan to study his work carefully.

Michael Atiyah is more than 89 years old. Most people are dead by that age – and almost all the remaining ones are hopelessly senile. But Michael Atiyah is a top mind. He has also co-written a paper about M-theory with Witten which I sort of liked. It's a guy who is close to string theory and I believe that this kind of background is appropriate for the Riemann Hypothesis.

It may be lots of fun on Monday morning. Stay tuned. And return to The Heidelberg Laureate Forum YouTube channel where the videos will be posted after the event. The live video hyperlink is at the top of the blog post.

Update Monday noon: I am almost sure the proof is hopelessly wrong.

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