Inter-universal Teichmüller theory and the ABC Conjecture

For this blog, I have been steering away from my professional work, but the on-going drama surrounding Shinichi Mochizuki and Peter Scholze is so interesting and has derailed so out of control it’s shedding light on an aspect of my sexual philosophical weltanschauung I feel compelled to bring the topic to my audience here.

So, since the technical nature of the works involved, a little quick introduction is necessary so you can follow the drama.

Shinichi Mochizuki entered Princeton University as an undergrad at the age of 16, eventually obtaining his Ph.D. in mathematics, also at Princeton, at the age of 23, and is considered an eminent mathematician in the field of anabelian geometry, which is actually completely beyond my ability to comprehend. Anabelian geometry is a subfield of algebraic number theory, which is by far the most difficult field in all of pure mathematics. This is the field where you encounter century old problems that have baffled the smartest people who ever lived for hundreds of years; things like Fermat’s Last Theorem, Jacobi Conjecture, Goldbach Conjecture, Twin Primes Conjecture, etc. all live in this field.

One of those famous conjectures involved in this controversy is the so-called ABC conjecture. It’s easy enough to actually describe what it does, as all those conjectures are, because they involve just real numbers, and their seeming simplicity absolutely belies the immense difficulties of ever trying to prove them. Having a Ph.D. in math at this point (I assume most of my readers here have yet to defend their Ph.D. thesis in math, which is the bare minimum requirement to even start discussing the work) doesn’t actually help you, in this case, to start comprehending the works behind those innocent-looking statements. And in the case of Mochizuki’s alleged proof, even being a Fields medalist (the Nobel prize equivalent for mathematics) is not sufficient.

Which, at this point, I would like to introduce our next math prodigy: Peter Scholze, Fields Medal recipient, the youngest tenured professor ever from University of Bonn in Germany, working in the field of arithmetic geometry. Receiving his Ph.D. in math at the age of 25. He was also a three times International Math Olympiad gold medalist when he was in high school and is considered one of the most talented mathematicians alive today, on par with other math prodigies such as Terence Tao and Jacob Lurie.

Okay, so now that the main characters are here, I can finally start telling you the drama that is unfolding and it’s honestly feeling like watching the remake of Newton vs. Leibniz.

What happened was Mochizuki invented Inter-universal Teichmüller theory (IUT), a whole new system of math that, if true, will be able to not just solve the ABC conjecture, but also a wide range of other unsolved conjectures in algebraic number theory. The preprints of IUT were almost 500 pages and were extremely abstruse and only a handful people in the world have the capability to understand them.

This was in 2012.

Recently, in 2020, Sholze and another colleague Jacob Stix claimed that after years of study, they have actually found an irreparable flaw in the logic in one of the IUT corollaries, and there was no foreseeable way to fill the gap. And apparently, from what I have read, by authors who were familiar with the work, because the way IUT is set up, there is actually no intermediate way to test the entire theory, and so if it can’t prove ABC, then it’s useless.

Scholze’s argument:

It’s impossible for me to understand the work of Mochizuki. Nobody in the whole world does other than a handful of people, but Scholze, who has studied IUT for several years, was actually able to simplify the works enough to a point that I was actually able to grasp some ideas from it, and this is what it boils down to:

Basically the IUT claims that hyperbolic curves can be completely characterized by their fundamental groups and we only need to study those fundamental groups. Which point Scholze disagrees and points to that actions on fundamental groups performed are not always commutative and Mochizuki had no way to fix it.

There is in fact a passage back and forth between (according to another theorem in IUT) hyperbolic curves and their fundamental groups via isomorphisms and it’s supposed to make the data more precise. Already, do you see a contradiction in IUT? If fundamental groups completely determine hyperbolic curve, then why does passage back and forth somehow improve the identification?

You can read Sholze’s original papers in the reference below.

This, apparently, has infuriated Mochizuki, who insisted that IUT is correct, and that the two mathematicians from Germany, Sholze and Stix are, and I quote, “profoundly ignorant of the actual mathematical content of inter-universal Teichmuller theory …” [See reference below]

He’s calling the two of most prominent mathematicians of our age imbeciles, and this is all in print.

And of course this is not over. Mochizuki had gone ahead and published his papers in a Japanese math journal, for which he is the chief editor, and in the publication, he claims to have solved the ABC conjecture. So in Japan, ABC conjecture is solved. The rest of the math community disagrees.

Then, in 2023, another mathematician Kirti Joshi from Arizona State University, who had studied Mochizuki’s papers, published a preprint that basically states that he was able to fill the gap in the original IUT and prove ABC conjecture.

However, Sholze disagrees and once again points out a gap in his proof and this happened in March of 2024, this year!

So now, instead of one, we have two flawed proofs of ABC conjecture. But that is not all, Joshi had reached out to Mochizuki to collaborate on IUT and this is what Mochizuki says about Joshi, who, while not as eminent a mathematician, is an associate professor of mathematics and has published numerous papers in credited math journals, “First of all, there was an entirely unanimous consensus that Joshi’s series of preprints was obviously mathematically meaningless, and that it was obvious that he did not have any idea what he was talking about.”

One mathematician has described this type of rhetoric as “simply unacceptable” in a professional setting.

But in academia, especially in math and physics, it seems that words like “you are either not listening or not understanding,” “You don’t seem to know what you are talking about at all,” “do you even understand?” are being thrown around all the time. And this is among people who have all obtained their Ph.D.’s in math or physics, people who are qualified to be doing research in their fields. Math and physics are very daunting fields of endeavor, and while I do not want to intimidate you, I do have to say that it’s not for everyone.

Not only that! But now Mochizuki has set up a foundation where he will award 1 million dollars to anyone who can find a flaw in his IUT! This is all very strange, and it’s honestly kind of embarrassing at this point. Also, there are emails where Mochizuki demand that Scholze and Stix retract their papers regarding finding flaws in IUT.

As Scholze said, the entire debacle is a great embarrassment for the mathematical community.

Let me end by saying that I wished that Mochizuki had focused his manuscript only on this mathematical point, and that I feel deeply sorry for Joshi for the rest of the manuscript (and more generally, I feel deep embarrassment as a member of the mathematical community.)

Honestly to me it’s like watching the battle of giants in real time, the gigantomachy of the smartest people alive on earth.

But I think the whole reason why I posted this here is to show you just how Japanese—East Asian societies and East Asian men in general—behave. Of course not all Japanese are this bad, but I’m not surprised that this happened in Japan. Japanese may be bad. Chinese is no better. A Chinese mathematician (Shing-Tung Yau of Harvard) tried to steal the proof of Poincare Conjecture from Grigori Perelman back in 2005.

This little anecdote just reinforces what I have always known—

The Fundamental Theorem of Jennifer Suzuki:

Seeking truth for truth’s sake was, is, and will after all always be a white men’s endeavor, and the greatest geniuses were, are and will always remain to be–white men.

Corollary:

If truth were an Asian woman (Nietzsche, 1898), then she will always choose to give herself to white man, and white man only.

Reference:

Click to access szpirostillaconjecture.pdf

https://www.math.columbia.edu/~woit/wordpress/?cat=33

Click to access 2023-08%20Brief%20report%20on%20the%20current%20situation%20surrounding%20inter-universal%20Teichmuller%20theory%20(IUT).pdf

Click to access reopen-e-mail-2022-06-30.pdf

Click to access Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf

https://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html

Click to access 2403.10430.pdf

https://mathoverflow.net/questions/467696/global-character-of-abc-szpiro-inequalities

https://mathoverflow.net/questions/468079/is-there-a-mistake-in-mochizukis-proof-of-theorem-1-10-in-iutt-iv