Author: Kevin Hartnett 2021-9-9 Translator: zzllrr Xiaole 2021-9-19
Michael Freedman's 1981 important proof of the four-dimensional Poincaré conjecture is on the verge of being lost. The editors of a new book are trying to save it.
One of the most important mathematical knowledge is on the verge of disappearing, perhaps forever. Now, a new book hopes to save it.
The Disc Embedding Theorem is a proof that Michael Freedman rewrote in 1981 — about the infinite network of disks — after years of lonely toil on the California coast. Friedman's proof answered a question that was one of the most important unsolved problems in mathematics at the time and a defining problem in Friedman's domain topology.
Friedman's proof feels amazing. No one at the time believed it could work — until Friedman personally persuaded some of the most respected people in the field. But while he won the favor of his contemporaries, written proofs are so full of holes and omissions that unless you have Friedman or someone who learned proof from him standing on your shoulders and guiding you, you can't keep up with its logic.
"I may not have taken the elaboration of written material as carefully as I should have," said Friedman, who today leads a Microsoft research group at the University of California, Santa Barbara, focused on building quantum computers.
Thus, the miracles that Friedman proved have disappeared into mythology.
Today, few mathematicians understand what he did, and those who know him are gradually withdrawing from the field. The result was that the studies involving his proof withered. Almost no one got the main result, and some mathematicians even questioned whether it was entirely correct.
In a 2012 MathOverflow post, one reviewer called the proof "a thesis monster" and said he "has never seen a mathematician who can make me believe he understands Friedman's proof." ”
The new book is the best effort to resolve this situation. It's the result of a collaboration of five young researchers who were captivated by the beauty of Freedman's proof and wanted to give it new life. It is nearly 500 pages long and illustrates the steps of Friedman's argument in complete detail using clear, consistent terminology. The goal is to turn this important but incomprehensible mathematics into something that an motivated undergraduate can learn in a semester.
"There's nothing left for imagination," says Arunima Ray of the Max Planck Institute for Mathematics in Bonn, who co-edited the book with Stefan Behrens of Bielefeld University, Boldizsár Kalmár of budapest University of Economics and Technology, Hoon Kim of Chonnam National University in South Korea, and Mark Powell of Durham University in the UNITED. "Everything is already determined."
Sort the spheres
In 1974, michael Freedman was 23 years old, and he looked at one of the biggest problems in topology. Topology is the field of mathematics that studies the fundamental characteristics of a space or manifold, as mathematicians call it.
It is known as the Poincaré conjecture, named after the French mathematician Henri Poincaré, who proposed it in 1904. Poincaré predicted that any shape or manifold with certain generic features would have to be equivalent or identical to a sphere. (Two manifolds are embryonic, when you can map all points on one manifold to points on another, while maintaining relative distances between points so that points that are close on the first manifold are still close together on the second manifold.) ）
Poincaré specifically considered three-dimensional manifolds, but mathematicians continued to consider manifolds of all dimensions. They also wanted to know if the conjecture would work for both manifolds. The first, called a "smooth" manifold, has no features such as sharp corners and allows you to perform calculus at each point. The second, known as "topological" manifolds, can have corners where calculus cannot be performed.
By the time Friedman began working on the problem, mathematicians had made considerable progress on the conjecture, including solving its topological version in 5 dimensions and beyond.
Friedman focused on the four-dimensional topology conjecture. It states that any topological manifold is a four-dimensional "homotopian" sphere, which is loosely equivalent to a four-dimensional sphere and is in fact homomorphic (strongly equivalent) to a four-dimensional sphere.
"The question we're asking is, is there a difference between these two equivalent concepts [for the tetrasphere]?" Ray said.
The four-dimensional version is arguably the most difficult version of the Poincaré problem. This is partly because the tools mathematicians use to solve higher-dimensional conjectures don't work in the more constrained settings of the four dimensions. (Another contender for the hardest version of the problem is the three-dimensional Poincaré conjecture, which was not resolved by Grigori Perelman until 2002.) ）
When Friedman started working, no one had any mature ideas about how to solve this problem — meaning that if he was to succeed, he would have to invent entirely new mathematics.
Important curves
Before diving into how he proves the Poincaré conjecture, it is worth delving into the true meaning of the problem.
A four-dimensional homotope sphere can be characterized by the way it interacts with curves drawn inside: interactions tell you some basic information about the larger space in which they interact.
In the four-dimensional case, these curves will be two-dimensional planes (typically, these curves are at most half of the larger spatial dimensions they draw internally). To understand the basic setup, consider a simpler example that involves one-dimensional curves that intersect in two-dimensional space, as follows:
These curves have something called algebraic intersection numbers. To calculate this number, calculate it from left to right and assign a -1 to each arc where they intersect ascent, and a +1 to each arc where they intersect. In this example, the leftmost intersection gets -1 and the rightmost intersection gets +1. Add them together and you will get the number of algebraic intersections of these two curves: 0.
The characteristic of the homotope sphere is that any pair of half-dimensional curves drawn inside it have an algebraic intersection of 0.
The same is true for regular spheres. But regular spheres also have a slightly different property associated with intersection: you can always draw two curves so that they don't intersect each other. Thus, homotopic spheres have the property that the number of algebraic intersections of a pair of curves is always 0, while regular spheres have the property that any pair of curves can be separated from each other, so that their number of geometric intersections is 0. That is, they don't actually intersect at all.
In order to prove the four-dimensional Poincaré conjecture, Friedman needed to prove that it was always possible to replace specific pairs of curves with number intersections of 0 and "push them away" from each other so that their geometric intersections remained 0. If you have pairs of curves where the algebraic intersection is 0, and you prove that you can always separate them, you prove that the spaces in which they are embedded must be regular spheres.
"It's like social distancing for these semi-dimensional manifolds," Ray said.
Previous work on the later dimensional version of the problem has established a way to do this. It involves looking for objects called Whitney discs, which are flat two-dimensional spaces surrounded by curves you want to separate.
These disks become a guide to a mathematical process called isotopes, in which you can move two curves away from each other. The presence of these flat Whitney discs ensures that the arc curve can be gradually moved downwards. When you do this, the disc begins to disappear, like the setting sun. Eventually, the disc disappears completely and the curves are separated.
"The Whitney disc gives you a path to isotopy. You keep moving one curve until the two curves are separated. The disc is like a roadmap for this process," Ray said.
When Friedman was confronted with the four-dimensional Poincaré conjecture, his main task was to prove that these flat Whitney disks existed when you had a pair of intersecting curves with an algebraic intersection of 0. Determining it was real took Friedman to unimaginably new heights in mathematics.
Undo the disc
In Friedman's work, he encountered a peculiar stumbling block that appeared on four dimensions. He needed to prove that it was always possible to separate intersecting two-dimensional curves—pushing them away from each other—and for that he had to establish the existence of the Whitney disk to ensure that separation was possible.
The problem is that in four dimensions, two-dimensional Whitney discs can cross each other rather than lie flat. The position where the disk intersects itself hinders the process of one curve sliding down from another. You can think of self-intersection as an obstacle, and when you try to pull it away from another curve, it grabs one of your curves.
"The disc was supposed to help me, but it turns out that the disc also intersects with itself," Ray said.
Therefore, Friedman needed to prove that it was always possible to cancel the position where the Whitney disc intersected, lay them flat, and then continue to separate. Luckily for him, he doesn't start from scratch. In the 1970s, a mathematician named Andrew Casson proposed a strategy to eliminate disc self-inbring.
The focus of the disc is to determine that the curves can be separated so that they do not intersect. If a disc itself contains an intersection, the way to mitigate it is the same: look for the second disc bounded by the intersecting part of the first disc. If you find the second disc, you know that you can eliminate the intersection in the first disc.
Okay, but what if the second disc (helping the first one) also intersects with itself? Then you look for the third disc contained in the second one. However, that disc may also intersect with itself, so you look for a fourth disc, and this process goes on and on, producing infinite stacks of discs within the disc—all the discs are erected, hoping to always be sure that the original disc is at the bottom so that it doesn't intersect.
Casson determined that these "Casson handles" roughly corresponded to actual Whitney disks—more precisely, homotopy equivalence—and he used this equivalence to study many important problems in four-dimensional topologies. But he couldn't prove that Carson's handles were equivalent to discs in a stronger sense—they were embryonic to discs. This stronger equivalence is exactly what mathematicians need in order to use the handle to prove the biggest unresolved problem.
"If we prove that these are truly honest disks, we can prove the Poincaré conjecture and a whole bunch of other things in the fourth dimension," Ray said. "But [Casson] can't do it."
Friedman's insight
From 1974 to 1981, Friedman spent seven years doing it. For most of the time, he barely talked to anyone about his work, except for his older colleague Robert Edwards, who was a mentor.
"He shut himself up [in San Diego] for seven years thinking about it. Peter Teichner of the Max Planck Institute for Mathematics said: "He didn't interact much with other people in the process of figuring it out. ”
Robion Kirby, now at the University of California, Berkeley, was one of the first mathematicians to learn about the Freedman proof. To assess the importance of the primary mathematical result, Kirby tried to imagine how long it would take before anyone else proposed it, and by that standard, Friedman's proof was the most astonishing result Kirby had seen in his long career.
"If he hadn't done it, I can't imagine who could do it, I don't know how long," Kirby said.
Friedman needed to prove that the Carson handle was equivalent to a flat Whitney disc: if you had a Carson handle, you had a Whitney disc, if you had a Whitney disc, you could separate the curves, and if you could separate the curves, you'd already determined that the homonymous ball was identical to the actual ball.
His strategy was to prove that you could build two objects out of the same set of parts— a Carson handle and a flat Whitney disc. The idea is that if you can build two things in the same part, they have to be equivalent in some sense. Freedman began the construction process and made considerable progress: he could build almost all casson handles and almost all discs with the same components.
But there are places where he can't complete the picture — like when he's creating a portrait, he can't see certain aspects of the subject's face. His final step, then, is to prove that those blank spaces in his photographs — places he can't see — don't matter from the point of view of the equivalent type he's after. That is, gaps in the picture are unlikely to prevent Casson handles from being homeomorphic to the disc, no matter what they contain.
"I have two puzzles and 99 out of 100 pieces match. Did all this leftover really change my space? Friedman proved they weren't," Ray said.
To accomplish this final step, Friedman borrowed a technique known in mathematics as bing topology, named after mathematician R.H. Bing, who developed it in the 1940s and 1950s. But he applied them to a whole new environment and came to a seemingly absurd conclusion — in the end, the gap didn't matter.
Kirby said: "That's what proves so compelling and makes it unlikely that others will find it." ”
Friedman completed his proof outline in the summer of 1981. Soon after, the factors that eventually put it in mathematical memory became apparent.
Spread the news
In August of that year, Friedman announced his proof at a small conference at the University of California, San Diego. About 10 of the most respected mathematicians attended the conference, who were most likely to learn about Friedman's work.
Before the event began, he sent out a 20-page copy of the handwritten manuscript outlining his proof. On the second night of the conference, Friedman began to display his work. He couldn't finish it at once, so his conversation continued until the next night. When he finished, his small audience was confused—Friedman's mentor Edwards was one of them. In a 2019 interview with the process, Edwards recalled the shock and disbelief of getting Friedman's speech.
"I think it's fair to say that everyone in the audience found his speech unbelievable and incomprehensible and thought his ideas were stupid and crazy," Edwards said.
Friedman's proof largely seems unlikely because it is not really fleshed out. He had an idea of how the proof should work, and had a strong, almost supernatural intuition that it would work. But he didn't really do that.
"I can't imagine how Mike would have the courage to announce a proof when he was so wobbly in the details," said Kirby, who also attended the meeting.
But later, several mathematicians stayed behind to talk to Friedman. At the very least, the importance of the potential outcome seems worthwhile. After two days of conversation, Edwards knew enough about what Friedman was trying to do to assess whether it really worked. On the first Saturday morning after the meeting, he realized it was.
"[Edwards] said, 'I was the first to really know that this was true,'" Kirby said.
Once Edwards is persuaded, he helps persuade others. In a way, that's enough. No higher mathematical committee has officially proven the results correct. The actual process of accepting new statements is more informal and relies on the consent of members of the mathematical community who should know best.
"Truth in mathematics means you convince the expert that your proof is correct. Then it becomes a reality," Tessina said. "Friedman persuaded all the experts that his proof was correct."
But this in itself is not enough to publish the results through this area. To do so, Friedman needed a written statement proving that people who had never met him could read and learn on their own. And that's something he never did.
Go on
Friedman submitted his proof outline to the Journal of Differential Geometry—that's all he really had. The magazine's editor, Chengtong Yau, assigns it to external experts for review before deciding whether to publish it — a standard guarantee for all academic publications. But the man he assigned it to was hardly an objective expert: Robert Edwards.
The review will still take time. The proof itself is 50 pages long, and Edwards finds that he is writing a dense mathematical note for each page of the proof. As the weeks passed, the journal's editors became restless. Edwards regularly receives calls from the journal's secretary asking if he has made a judgment on the legitimacy of the proof. In the same interview in 2019, Edwards explained that, in the end, he told the magazine that the proof was correct, even though he knew he didn't have time to fully check it out.
"The next time the secretary called me, I said', 'Yes, the paper is correct, I assure you.' But I can't generate a proper referee report anytime soon. 'So they decided to accept it and release it as is,' he said.
This paper was published in 1982. It contains typos and spelling mistakes and is actually still the outline that Friedman distributed after he finished his work. Anyone trying to read it will need to fill out the many steps of this entirely new argument themselves.
The limitations of the published articles were immediately apparent, but no one stepped forward to address them. Friedman turned to other work and no longer taught his Poincaré proof. Almost a decade later, in 1990, a book appeared that tried to provide a more understandable version of the proof. It was written by Friedman and Frank Quinn and is now at Virginia Tech and State University, though it was written primarily by Quinn.
The version of the book is barely readable. It assumes that the reader brings a certain amount of background knowledge to the book that almost no one actually possesses. There is no way to read it and learn the proof from scratch.
"If you're lucky enough to be with someone who understands proof, you can still learn it," Teichner said. "But people who come back to [written] sources realize they can't."
For decades, this has been where things have always been: one of the most astonishing results in the history of mathematics is known to a few, while others are not.
Others in the mathematical community may have moved on like Friedman, but his proof is too important to ignore entirely. As a result, the community adapted to this strange situation. Many researchers have used Friedman's proof as a black box. If you assume that his proof is correct, you can prove a lot of other theorems about four-dimensional manifolds, and many mathematicians have done it.
Powell said: "If you just accept that it is real, you can use it in many ways." "But that doesn't mean you want to accept everything with faith."
Over time, as younger researchers enter the field of mathematics and have the option to work in whatever field they want, fewer and fewer people choose to prove their studies.
Friedman understood. "Working in an area where you don't understand fundamental theorems isn't that satisfying," he said. "Basically, people under the age of 40 don't know that a proof of the situation has occurred, and this bit of information may eventually be lost, which is a bit scary."
It was at this point that Tessina— who learned the proof from Friedman himself in the early 1990s — decided to launch a rescue mission. He wanted to create a text so that anyone who was qualified could learn to prove it on their own.
"I decided it was time to write something you could understand," he said.
Future Proof friedman
Tessina first went straight back to the source. In 2013, he asked Friedman to give a series of lectures over a semester at the Max Planck Institute, describing proof — a modern version of a speech he gave 30 years earlier — to announce the results. Friedman eagerly agreed.
"He must have feared it would be lost. That's why he's so supportive," Teichner said.
Back in 1981, Friedman lectured to several veterans in the field— he needed to win over the experts. This time, his audience was 50 young mathematicians gathered by Tessina to take over the baton. Friedman's lectures delivered via video from his office in Santa Barbara were a major event in the topological world.
"At my institution, we used to have Friedman Lectures on Friday afternoons, where we would drink beer and watch him talk about his proof," said Ray, who was a graduate student at Rice University in Houston at the time.
After the lecture, mathematician Stefan Behrens worked to translate Friedman's comments into more formal lecture notes. A few years later, in 2016, Powell and other mathematicians, including Behrens, published a new series of lectures based on these notes, continuing to translate Friedman's work into something more enduring.
"Mark lectures, we start filling in more and more details in these handouts, and then we just go from there," Ray said.
Over the next five years, Powell, Ray, and their three co-editors organized a team of mathematicians to turn Friedman's proof into a book. The final product was released in July with nearly 500 pages and including contributions from 20 different authors. Friedman hopes the book will reinvigorate his revolutionary field of mathematics.
Mark Powell and Arunima Ray created a new book-length version of the Freedman proof because they wanted to understand it for themselves — and share it with a new generation of mathematicians.
Courtesy of Victoria Greener; Stephen Friedel
"I think the book came at the right time. People are looking at four-dimensional manifolds in a whole new light," he said.
The book improves on friedman's written statements in several ways. While writing the book, the authors discovered some errors in the arguments Friedman used to prove different theorems in the original journal articles. This book fixes these. It also provides a comprehensive introduction to Bing topology, a mathematical area that Friedman used to prove that the gap between his Carson handle and the Whitney disc structure was irrelevant. All in all, the book is designed to be pedagogical and easy to understand. The previous chapters provide an outline of the proof, and the later chapters fill in the details.
"There's a summary first, then a more detailed summary, then a full detail, which should make it readable," Powell said. "Before you get all the details, you can get a big picture of what's going to happen. But we still have all the details. ”
Editors want to push Friedman's powerful technique back into the mainstream of mathematical thinking. The third part of the book details the biggest open problems in four-dimensional topologies that researchers might be able to deal with once they have Friedman's knowledge of proof.
"This part of the book has nothing to do with the proof of Friedman's original work," Ray said. "It talks about how to use it to do what comes next."
And, several mathematicians involved in the book have already conducted new research based on Friedman's ideas. A paper published in 2013 at the beginning of the book's process uncovered some new uses for techniques that were previously dormant in bing topologies. Another, from last year, used ideas the editors learned while assembling the book to solve the problem of "surgery" on knots in a four-dimensional manifold.
"It's moving forward now because they're used to using the disc embedding theorem," Teichner said.
This book has played a supporting role in the field of mathematics, and may even be indispensable. But the editors say their motivation isn't just to achieve the actual purpose of the long-term project. When they started working, Friedman's proof was beautiful, but hidden. Now, it's finally on full display.
Correction: September 10, 2021
Friedman announced his certification at the University of California, San Diego, not at the University of San Diego. The article has been modified accordingly. The graph depicting intersecting curves has also been modified to more accurately reflect the content of the article.