Titans from mathematics have clashed over the epic evidence of the abc hypothesis

 3r33333. 3r3-31.
Two mathematicians claim to have found a hole in the very heart of the evidence that has been shaking the mathematical community of 3r-33r for six years. 3r33333.  3r33333. Titans from mathematics have clashed over the epic evidence of the abc hypothesis , published in September 2018 on the Internet, 3r3133. Peter Scholze
from the University of Bonn and 3—3–315. Jacob Stix
from Goethe University in Frankfurt described what Styx calls a “serious and irreplaceable gap” in 3r317. tremendous
series. volume works Shiniti Mochizuki , the famous genius mathematician from Kyoto University. Published in the Internet in 201? Mochizuki allegedly prove 3r-327. abc hypothesis
, one of the most far-reaching tasks in number theory . 3r33333.  3r33333. 3r3333.
3r33333.  3r33333. Despite the many conferences that attempted to explain Mochizuki’s evidence, number theory specialists struggled with the underlying ideas. His series of works with a total volume of more than 500 pages are written in an obscure style, and refer to his previous work of about 500 pages, which leads to the appearance of a “feeling of infinite regress”, as 3r3209. put it
mathematician Brian Conrad from Stanford University. 3r33333.  3r33333. 3r33333.  3r33333. From those who studied evidence, mathematicians believe that 12 to 18 people are correct, as wrote to me. Ivan Fesenko from the University of Nottingham by email. But as commented on the situation in the discussion of the evidence in a blog last December, Konrad, for the fidelity of the evidence was entrusted only to mathematicians from the "Mochizuki inner circle." "There is no longer anyone willing to declare, even unofficially, the confidence in the completeness of the evidence." 3r33333.  3r33333. 3r33333.  3r33333. However, as wrote r3r3297. in his blog 3r3355. Frank Kalegari from the University of Chicago in December, "mathematicians are reluctant to report problems with the proof of Mochizuki, since they cannot point out a specific error." 3r33333.  3r33333. 3r33333.  3r33333. Now everything has changed. In their report, Scholze and Stix argue that the line of reasoning near the end of the proof of “Conclusion ???” in the third of the four works of Mochizuki is fundamentally erroneous. And this conclusion is necessary for the proposed evidence of the abc-hypothesis. 3r33333.  3r33333. 3r33333.  3r33333. “It seems to me that the issue with the abc hypothesis remains open,” said Scholze. “And every person has a chance to prove it.” 3r33333.  3r33333. 3r33333.  3r33333. Peter Scholze [/i] 3r33333.  3r33333. 3r33333.  3r33333. The conclusions of Scholze and Stix are based not only on their own study of the work, but also on the weekly visit they put on Mochizuki and his colleague Yuichiro Hoshi in March at Kyoto University, which was held to discuss this evidence. Scholze says that this visit helped him and Styx tremendously to get to the core of their objections. As a result, a couple of scientists "came to the conclusion that there was no evidence," they write in the report. 3r33333.  3r33333. 3r33333.  3r33333. However, this meeting ended in dissatisfaction with the parties. Mochizuki could not convince Scholze and Stix that his proof was correct, but they could not convince him that it was wrong. Mochizuki already posted the Scholze and Stix report on his website, and added 3r382 to them. Some of their objections
. 3r33333.  3r33333. 3r33333.  3r33333. In them, Mochizuki attributed the criticism of Scholze and Stix to the account of "certain fundamental misinterpretations" of his work. Their “negative attitude,” he writes, “does not indicate the presence of any flaws” in his theory. 3r33333.  3r33333. 3r33333.  3r33333. Just as Mochizuki’s serious reputation led mathematicians to view his work as a serious attempt at proving a hypothesis, Scholz and Stix’s reputation ensures that mathematicians pay attention to what they want to say. Scholze, although he was only 30 years old, quickly rose to the top in his field. In August, he received Fields Award 3–3–3297. , the highest reward in mathematics. Styx is an expert in Mochizuki research, anabelian geometry. 3r33333.  3r33333. 3r33333.  3r33333. “Peter and Jacob are extremely cautious and thoughtful mathematicians,” said Konrad. “If they have any concerns, they really should be clarified.” 3r33333.  3r33333. 3r33333.  3r33333.

The stumbling block 3r3204. 3r33333.  3r33333. The abc hypothesis, which Konrad called "one of the most prominent hypotheses in number theory," begins with one of the simplest equations that you can imagine: a + b = c. Three numbers a, b and c are positive integers that have no common prime divisors. That is, we can consider the equation 8 + 9 = 17 or 5 + 16 = 2? but not 6 + 9 = 1? since the numbers ? 9 and 15 are divided by 3.
 3r33333. 3r33333.  3r33333. Taking such an equation, we can consider all the prime numbers into which any of the three numbers involved in the equation are divided — for example, in the case of the equation 5 + 16 = 2? these prime numbers will be ? ? 5 and 7. Their product will be 210 , and it is much larger than any of the numbers involved in the equation. Conversely, in the equation 5 + 27 = 3? the prime numbers ? 3 and 5 are involved, the product of which is equal to 30 - and this is less than the number 32 participating in the equation. The product is so small, because the numbers 27 and 32 have very small simple divisors (3 and 2), which are simply repeated many times to get these numbers. 3r33333.  3r33333. 3r33333.  3r33333. If you start playing with other abc triples, you may find that this second option is extremely rare. For example, among 3044 different triples, in which the members of a and b are less than 10? there are only seven, where the product of prime divisors is less than c. The abc hypothesis formulated in the 1980s formalizes the intuitive idea of ​​the rarity of such triples. 3r33333.  3r33333. 3r33333.  3r33333. Returning to the example 5 + 27 = 32. 32 is more than 3? but not much. This is less than 30 r3r3150. 2 [/sup] , or 30 r3r3150. ???r3r3151. , or even 30 r3r3150. ???r3151. equal to ???. The abc hypothesis says that if you choose any degree greater than ? then there will be only a finite number of abc triples, which c will have more products of prime divisors raised to the chosen degree. 3r33333.  3r33333. 3r33333.  3r33333. “The abc hypothesis is a very simple statement regarding multiplication and division,” said 3r3128. Minyun Kim
from the University of Oxford. He said that with such a statement, "there is a feeling that you are revealing some very fundamental structure of numerical systems that you have not seen before." 3r33333.  3r33333. 3r33333.  3r33333. The simplicity of the equation a + b = c means that a wide range of other problems fall under its influence. For example, 3r3134. Fermat's great theorem
associated with equations of the form x n + y n = z n , and 3r3142. Catalan's hypothesis
, arguing that 8 and 9 are the only consecutive two perfect powers of[чисел, выражающихся целым числом в целой степени /прим. перев.](since 8 = 2 3 and 9 = 3 3r3-33150. 2 [/sup] ), speaks of an equation of the form x m + 1 = y n . The abc hypothesis (in a certain form) would give new proofs to these two theorems and solve a whole mountain of open problems related to it. 3r33333.  3r33333. 3r33333.  3r33333. Jacob Stix [/i] 3r33333.  3r33333. 3r33333.  3r33333. This hypothesis "as if all the time is on the border between the known and the unknown" - wrote r3r3297. 3r3167. Dorian Goldfeld from Columbia University. 3r33333.  3r33333. 3r33333.  3r33333. The scale of the consequences of proving the hypothesis convinced the number theory specialists that it would be very difficult to prove it. Therefore, when information spread in 2012 that Mochizuki presented evidence, many mathematicians plunged into his work with ecstasy - but only to be stumped because of an unfamiliar language and an unusual presentation of information. The definitions stretched over several pages, followed by theorems with equally long statements, and their proofs were described with phrases like “immediately follows from the definition”. 3r33333.  3r33333. 3r33333.  3r33333. “Every time I hear about the analysis of the work of Mochizuki done by an expert (unofficial), his review turns out to be outrageously familiar: the wide fields of trivial things, followed by huge mountains of unjustified conclusions,” - 3r3252. wrote r3r3297. Calegari on his blog in December. 3r33333.  3r33333. 3r33333.  3r33333. Scholze was one of the first readers of the work. He is known for being able to quickly absorb mathematics, penetrating deep into it, so he moved further than many theorists, and finished what he called the “rough reading” of four major works shortly after their appearance. Scholze was embarrassed by long theorems with short proofs that seemed true to him, but unfounded. Later he wrote r3r3297. that in the two intermediate works "little is happening." 3r33333.  3r33333. 3r33333.  3r33333. Then Scholze reached conclusion ??? in the third paper. Mathematicians usually use the word "conclusion" to denote a theorem that is secondary to the previous, more important one. But in the case of the conclusion of ??? from Mochizuki, mathematicians agree that this is the main theorem for proving the abc hypothesis. Without it, “there is no proof —
wrote r3r3297. Calegari “This is a critical step.” 3r33333.  3r33333. 3r33333.  3r33333. This conclusion is the only theorem in two intermediate works, the proof of which takes more than a few lines — it stretches nine pages. Passing through them, Scholze reached the point at which he could no longer follow the logic. 3r33333.  3r33333. 3r33333.  3r33333. At that time he was only 2? and he believed that the proof was incorrect. But he practically did not enter into the discussion of the works, unless he was asked directly about them. After all, in the end, he thought, other mathematicians would surely find in these works meaningful ideas that he missed. Or perhaps they will eventually come to the same conclusion as he. One way or another, he thought, the mathematical community would be able to sort things out. 3r33333.  3r33333. 3r33333.  3r33333.
Escher's staircase

3r33333.  3r33333. In the meantime, other mathematicians struggled with impassable jobs. Many had high hopes for
dedicated to the work of Mochizuki, scheduled for the end of 2015 at Oxford University. But when several colleagues Mochizuki tried to explain the key ideas of the proof, a “cloud of fog” descended on the listeners, as Konrad wrote in a report shortly after the meeting. “It was necessary for people who understood this work to be more successfully explained to specialists in arithmetic geometry, which lies at its core” - 3r3209. wrote r3r3297. he. 3r33333.  3r33333. 3r33333.  3r33333. Within a few days after his post, Konrad received unexpected letters from three mathematicians (one of whom was Scholz), describing the same thing: they were able to read and understand the works until they reached a certain point. “Each of the three was stopped by evidence 3.1?” 3r33252. wrote r3r3297. later Konrad. 3r33333.  3r33333. 3r33333.  3r33333. Kim heard similar feedback on the conclusion of ??? and from another mathematician, Teruhisi Koshikawa, who works at Kyoto University. Styx also stumbled on this spot. Gradually, many number theory specialists learned that this conclusion was a stumbling block, but it was not clear whether there was a hole in his proof, or whether Mochizuki needed to explain his reasoning better. 3r33333.  3r33333. 3r33333.  3r33333. Then in 201? to the horror of many theorists, it was rumored that Mochizuki’s work had been accepted for publication. Mochizuki himself was the editor-in-chief of this magazine, 3r-3225. Publications of the Research Institute for Mathematical Sciences
. Kalegari called this situation "3r33252. Poor looking 3r-3297." (Although the editor in such situations is usually barred from making decisions). But most of all mathematicians were worried that the work was still unreadable. 3r33333.  3r33333. 3r33333.  3r33333. Shin-ichi Mochizuki on video at the 2015 conference on his proof of 3r-3237. 3r33333.  3r33333. 3r33333.  3r33333. “Not a single expert who claims to understand the evidence failed to explain it to any of the multitude of experts who remain confused.” - 3r3242. wrote r3r3297. Matthew Emerton from the University of Chicago. 3r33333.  3r33333. 3r33333.  3r33333. Calegari wrote an article describing this situation as "3r33252. a complete failure, 3r-3297.", and his point of view was picked up by eminent theorists. “We have a ridiculous situation in which abc is considered a theorem in Kyoto and a hypothesis in all other places,” Kalegari wrote. 3r33333.  3r33333. 3r33333.  3r33333. The PRIMS magazine soon responded to press inquiries with a statement in which it clarified that the works were not accepted for publication. However, even before that, Scholze decided to publicly state what he had already said in private conversations for a long time to many theorists. He decided that all this discussion of evidence was “too social.” “Everyone said that this evidence does not seem so, but no one said:“ There is a place where no one understood the evidence. ” 3r33333.  3r33333. 3r33333.  3r33333. In the comments on the post, Kalegari Scholze wrote that he “could not follow the logic after fig. 3.8 in proof of conclusion ???. He added that mathematicians, "who claim to understand the evidence, do not want to admit that they need to add something there." 3r33333.  3r33333. 3r33333.  3r33333. Shigefumi Mori , Mochizuki's colleague from Kyoto University, winner of the Fields Award, wrote Scholze with a proposal to organize a meeting with Mochizuki. Scholze, in turn, contacted Styx, and in March, the couple went to Kyoto to discuss the stumbling block in evidence with Mochizuki and Hoshi. 3r33333.  3r33333. 3r33333.  3r33333. Mochizuki's approach to the abc hypothesis takes the task to the 3r3-33272 domain. elliptic curves
, a special type of cubic equations with two variables, x and y. This transition, known even before Mochizuki, is simply carried out — you need to associate each abc-equation with an elliptic curve, whose graph intersects the x-axis at points a, b and at the origin — however, it allows mathematicians to use the rich structure of elliptic curves that combine number theory with geometry, integral numbering and other areas. (The same transition is at the center of the proof of the Great Fermat theorem of 199? performed 3r-3274. Andrew Wiles 3r-?397.). 3r33333.  3r33333. 3r33333.  3r33333. As a result, the abc hypothesis reduces to the proof of the inequality between two quantities associated with elliptic curves. Mochizuki's work translates this inequality into another form, which, as Styx said, can be represented as a comparison of the volumes of the two sets. In conclusion, ??? offers his own proof of this inequality, which, if true, would prove the abc hypothesis. 3r33333.  3r33333. 3r33333.  3r33333. In the proof, as described by Scholze and Stix, the volumes of the two sets are treated as if they are inside two different copies of real numbers, represented as hPart of a circle of six different copies of real numbers, as well as a markup, explaining how each copy is associated with its neighbor in a circle. To track the relationship of set volumes with each other, it is necessary to understand how measurements of volume in one copy relate to measurements in other copies, as Styx said. 3r33333.  3r33333. 3r33333.  3r33333. “If you have an inequality of two objects, but at the same time the measuring ruler is compressed several times beyond your control, then you lose control over what inequality generally means,” said Styx. 3r33333.  3r33333. 3r33333.  3r33333. Scholze and Styx believe that it is at this critical moment of the proof that everything collapses. In Mochizuki markup, the measuring lines are logically compatible with each other. But when you go round the circle, said Styx, you have a ruler, not like the one that will be if you go the other way. This situation, he said, resembles the famous closed staircase Escher on which you can climb, and then find yourself in the same place[правильнее сказать, что это лестница Пенроуза, по мотивам которой Эшер сделал известный рисунок /прим. перев.]. 3r33333.  3r33333. 3r33333.  3r33333. Scholze and Styx concluded that this incompatibility of volume measurements means that the wrong values ​​are compared in the final inequality. And if you correct everything so that the volumes become comparable, then the inequality becomes meaningless, they say. 3r33333.  3r33333. 3r33333.  3r33333. Scholze and Styx “found a way with which the proof doesn’t work,” said Kiran Kedlay, a mathematician at the University of California at San Diego, who studied in detail the work of Mochizuki. “So, if the proof is true, it should work with something else, with something less explicit,” than what Scholz and Styx describe. 3r33333.  3r33333. 3r33333.  3r33333. Mochizuki argues that this is the presence of something less obvious. He writes that Scholze and Styx are mistaken, arbitrarily equating mathematical objects that should be considered different. When he told colleagues about the essence of the objections of Scholze and Stix, he writes, his description “was met with remarkably general surprise and even distrust (and then also ridiculed) that such an incredible misunderstanding could have arisen at all.” 3r33333.  3r33333. 3r33333.  3r33333. Now mathematicians will have to digest the arguments of Scholze and Stix and the answer of Mochizuki. Scholze hopes that, unlike the situation with Mochizuki’s original work, this process will not last long, since their nature with the Styx objections is not so technically complex. Other theorists “should be able to easily follow the line of our discussion that we had with Mochizuki,” he said. 3r33333.  3r33333. 3r33333.  3r33333. Mochizuki, everything is completely different. From his point of view, criticism of Scholze and Stix comes from "lack of time to properly delve into the mathematics under discussion," which may be due to "a feeling of deep discomfort or unfamiliarity with the new way of thinking about familiar mathematical objects." 3r33333.  3r33333. 3r33333.  3r33333. Mathematicians, who were skeptical about the proof of Mochizuki before, may well decide that the Scholze and Stix report puts an end to this story, Kim said. Others will want to study the reports themselves, and this, Kim believes, has already begun. “I don’t think I’ll be able to avoid having to check everything myself before I decide something for myself,” he wrote by mail. 3r33333.  3r33333. 3r33333.  3r33333. Over the past few years, many number theory specialists have stopped trying to understand the work of Mochizuki. But if Mochizuki or his followers can provide a detailed and coherent explanation of why the picture of Scholze and Stix is ​​too simplified (if so), “it can do a lot to remove the fatigue associated with this issue and inspire people to try again” - said Kedlay. 3r33333.  3r33333. 3r33333.  3r33333. Meanwhile, Scholze says: “I think this cannot be considered evidence until Mochizuki makes a serious alteration and does not explain the key step much better.” He himself, in his words, “I do not see a key idea that could bring us closer to the proof of the abc hypothesis.” 3r33333.  3r33333. 3r33333.  3r33333. Regardless of the final outcome of the discussion, the clear designation of a specific place for the proof of Mochizuki should clarify everything very well, said Kim. “What Jacob and Peter succeeded in doing a very important service to the community,” he said. “Whatever happens, I’m sure these reports will be some kind of progress.” 3r33333. 3r33333. 3r33333. 3r33337. ! function (e) {function t (t, n) {if (! 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