Pyramid of people

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In the 1830s, the Irish mathematician William Rowan Hamilton reformulated Newton’s laws of motion, finding deep mathematical symmetries between an object’s position and its momentum. Then in the mid-1980s the mathematician Mikhail Gromov developed a set of techniques that transformed Hamilton’s idea into a full-blown area of mathematical research. Within a decade, mathematicians from a broad range of backgrounds had converged to explore the possibilities in a field that came to be known as “symplectic geometry.”

The result was something like the opening of a gold-rush town. People from many different areas of mathematics hurried to establish the field and lay claim to its fruits. Research developed rapidly, but without the shared background knowledge typically found in mature areas of mathematics. This made it hard for mathematicians to tell when new results were completely correct. By the start of the 21st century it was evident to close observers that significant errors had been built into the foundations of symplectic geometry.

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The field continued to grow, even as the errors went largely unaddressed. Symplectic geometers simply tried to cordon off the errors and prove what they could without addressing the foundational flaws. Yet the situation eventually became untenable. This was partly because symplectic geometry began to run out of problems that could be solved independently of the foundational issues, but also because, in 2012, a pair of researchers — Dusa McDuff, a prominent symplectic geometer at Barnard College and author of a pair of canonical textbooks in the field, and Katrin Wehrheim, a mathematician now at the University of California, Berkeley — began publishing papers that called attention to the problems, including some in McDuff’s own previous work. Most notably, they raised pointed questions about the accuracy of a difficult, important paper by Kenji Fukaya, a mathematician now at Stony Brook University, and his co-author, Kaoru Ono of Kyoto University, that was first posted in 1996.

This critique of Fukaya’s work — and the attention McDuff and Wehrheim have drawn to symplectic geometry’s shaky foundations in general — has created significant controversy in the field. Tensions arose between McDuff and Wehrheim on one side and Fukaya on the other about the seriousness of the errors in his work, and who should get credit for fixing them.

More broadly, the controversy highlights the uncomfortable nature of pointing out problems that many mathematicians preferred to ignore. “A lot of people sort of knew things weren’t right,” McDuff said, referring to errors in a number of important papers. “They can say, ‘It doesn’t really matter, things will work out, enough [of the foundation] is right, surely something is right.’ But when you got down to it, we couldn’t find anything that was absolutely right.”

The Orbit Counters

The field of symplectic geometry begins with the movement of particles in space. In flat, Euclidean space, that motion can be described in a straightforward way by Newton’s equations of motion. No further wrangling is required. In curved space like a sphere, a torus or the space-time we actually inhabit, the situation is more mathematically complicated.

This is the situation William Rowan Hamilton found himself considering as he studied classical mechanics in the early 19th century. If you think of a planet orbiting a star, there are several things you might want to know about its motion at a given point in time. One might be its position — where exactly it is in space. Another might be its momentum — how fast it’s moving and in what direction. The classical Newtonian approach considers these two values separately. But Hamilton realized that there is a way to write down equations that are equivalent to Newton’s laws of motion that put position and momentum on equal footing.

To see how that recasting works, think of the planet as moving along the curved surface of a sphere (which is not so different from the curved space-time along which the planet actually moves). Its position at any point in time can be described by two coordinate points equivalent to its longitude and latitude. Its momentum can be described as a vector, which is a line that is tangent to the sphere at a given position. If you consider all possible momentum vectors, you have a two-dimensional plane, which you can picture as balancing on top of the sphere and touching it precisely at the point of the planet’s location.

You could perform that same construction for every possible position on the surface of the sphere. So now you’d have a board balancing on each point of the sphere, which is a lot to keep track of. But there’s a simpler way to imagine this: You could combine all those boards (or “tangent spaces”) into a new geometric space. While each point on the original sphere had two coordinate values associated to it — its longitude and latitude — each point on this new geometric space has four coordinate values associated to it: the two coordinates for position plus two more coordinates that describe the planet’s momentum. In mathematical terms, this new shape, or manifold, is known as the “tangent bundle” of the original sphere. For technical reasons, it is more convenient to consider instead a nearly equivalent object called the “cotangent bundle.” This cotangent bundle can be thought of as the first symplectic manifold.

To understand Hamilton’s perspective on Newton’s laws, imagine, again, the planet whose position and momentum are represented by a point in this new geometric space. Hamilton developed a function, the Hamiltonian function, that takes in the position and momentum associated to the point and spits out another number, the object’s energy. This information can be used to create a “Hamiltonian vector field,” which tells you how the planet’s position and momentum evolve or “flow” over time.

Symplectic manifolds and Hamiltonian functions arose from physics, but beginning in the mid-1980s they took on a mathematical life of their own as abstract objects with no particular correspondence to anything in the world. Instead of the cotangent bundle of a two-dimensional sphere, you might have an eight-dimensional manifold. And instead of thinking about how physical characteristics like position and momentum change, you might just study how points in a symplectic manifold evolve over time while flowing along vector fields associated to any Hamiltonian function (not just those that correspond to some physical value like energy).

Lucy Reading-Ikkanda/Quanta Magazine

Once they were redefined as mathematical objects, it became possible to ask all sorts of interesting questions about the properties of symplectic manifolds and, in particular, the dynamics of Hamiltonian vector fields. For example, imagine a particle (or planet) that flows along the vector field and returns to where it started. Mathematicians call this a “closed orbit.”

You can get an intuitive sense of the significance of these closed orbits by imagining the surface of a badly warped table. You might learn something interesting about the nature of the table by counting the number of positions from which a marble, rolled from that position, circles back to its starting location. By asking questions about closed orbits, mathematicians can investigate the properties of a space more generally.

A closed orbit can also be thought of as a “fixed point” of a special kind of function called a symplectomorphism. In the 1980s the Russian mathematician Vladimir Arnold formalized the study of these fixed points in what is now called the Arnold conjecture. The conjecture predicts that these special functions have more fixed points than the broader class of functions studied in traditional topology. In this way, the Arnold conjecture called attention to the first, most fundamental difference between topological manifolds and symplectic manifolds: They have a more rigid structure.

The Arnold conjecture served as a major motivating problem in symplectic geometry — and proving it became the new field’s first major goal. Any successful proof would need to include a technique for counting fixed points. And that technique would also likely serve as a foundational tool in the field — one that future research would rely upon. Thus, the intense pursuit of a proof of the Arnold conjecture was entwined with the more workaday tasks of establishing the foundations of a new field of research. That entanglement created an uneasy combination of incentives — to work fast to claim a proof, but also to go slow to make sure the foundation was stable — that was to catch up with symplectic geometry years later.

How to Count to Infinity

In the 1990s the most promising strategy for counting fixed points on symplectic manifolds came from Kenji Fukaya, who was at Kyoto University at the time, and his collaborator, Kaoru Ono. At the time they released their approach, Fukaya was already an acclaimed mathematician: He’d given a prestigious invited talk at the 1990 International Congress of Mathematicians and had received a number of other awards for his fundamental contributions to different areas of geometry. He also had a reputation for publishing visionary approaches to mathematics before he’d worked out all the details.

Paul Guetter Photographs

Kenji Fukaya, a mathematician at Stony Brook University, argues that his work has always been both complete and correct.

“He would write a 120-page-long thing in the mid-1990s where he would explain a lot of very beautiful ideas, and in the end he would say, ‘We don’t quite have a complete proof for this fact,’” said Mohammed Abouzaid, a symplectic geometer at Columbia University. “This is very unusual for mathematicians, who tend to hoard their ideas and don’t want to show something which is not yet a polished gem.”

Fukaya and Ono saw the Arnold Conjecture as essentially a counting problem: What’s the best way to tally fixed points of symplectomorphisms on symplectic manifolds?

One method for tallying comes from work by the pioneering mathematician Andreas Floer and involves counting another complicated type of object called a “pseudo-holomorphic curve.” Counting these objects amounts to solving a geometry problem about the intersection points of two very complicated spaces. This method only works if all the intersections are clean cuts. To see the importance of having clean cuts for counting points of intersection, imagine you have the graph of a function and you want to count the number of points at which it intersects the x-axis. If the function passes through the x-axis cleanly at each intersection, the counting is easy. But if the function runs exactly along the x-axis for a stretch, the function and the x-axis now share an infinite number of intersection points. The intersection points of the two become literally impossible to count.

In situations where this happens, mathematicians fix the overlap by perturbing the function — adjusting it slightly. This has the effect of wiggling the graph of the function so that lines cross at a single point, achieving what mathematicians call “transversality.”

Lucy Reading-Ikkanda/Quanta Magazine

Fukaya and Ono were dealing with complicated functions on spaces that are far more tangled than the x-y plane, but the principle was the same. Achieving transversality under these conditions turned out to be a difficult task with a lot of technical nuance. “It became increasingly clear with Fukaya trying to prove the Arnold conjecture’s most general setup that it’s not always possible to achieve transversality by simple, naive methods,” said Yakov Eliashberg, a prominent symplectic geometer at Stanford University.

The main obstacle to making all intersections transversal was that it wasn’t possible to wiggle the entire graph of the function at once. So symplectic geometers had to find a way to cut the function space into many “local” regions, wiggle each region, and then add the intersections from each region to get an overall count.

“You have some horrible space and you want to perturb it a little so that you can get a finite number of things to count,” McDuff said. “You can perturb it locally, but somehow you have to fit together those perturbations in some consistent way. That’s a delicate problem, and I think the delicacy of that problem was not appreciated.”

In their 1996 paper, Fukaya and Ono stated that they used Floer’s method to solve this problem, and that they had achieved a complete proof of the Arnold conjecture. To obtain the proof — and overcome the obstacles around counting and transversality — they introduced a new mathematical object called Kuranishi structures. If Kuranishi structures worked, they belonged among the foundational techniques in symplectic geometry and would open up huge new areas of research.

But that’s not what happened. Instead the technique languished amid uncertainty in the mathematical community about whether Fukaya’s approach worked as completely as he said it did.

The End of the Low-Hanging Fruit

In mathematics, it takes a community to read a paper. At the time that Fukaya and Ono published their work on Kuranishi structures, symplectic geometry was still a loosely assembled collection of researchers from different mathematical backgrounds — algebraists, topologists, analysts — all interested in the same problems, but without a common language for discussing them.

In this environment, concepts that might have been clear and obvious to one mathematician weren’t necessarily so to others. Fukaya’s paper included an important reference to a paper from 1986. The reference was brief, but consequential for his argument, and hard to follow for anyone who didn’t already know that work.

“When you write a proof, it is implicitly checkable by somebody who has the same background as the person who wrote it, or at least sufficiently similar so that when they say, ‘You can easily see such a thing,’ well, you can easily see such a thing,” Abouzaid said. “But when you have a new subject, it’s difficult to figure out what is easy to see.”

Fukaya’s paper proved difficult to read. Rather than guiding future research, it got ignored. “There were people who tried to read it and they couldn’t, they had problems, so the adoption was actually extremely slow; it didn’t happen,” said Helmut Hofer, a mathematician at the Institute for Advanced Study in Princeton, New Jersey, who has been developing foundational techniques for symplectic geometry since the 1990s. “A lot of people just listened to other people and said, ‘If they have difficulties, I don’t even want to try.’”

Fukaya explains that in the years after he published his paper on Kuranishi structures, he did what he could to make his work intelligible. “We tried many things. I talked in many conferences, wrote many papers, abstract and expository, but none of it worked. We tried so many things.”

During the years that Fukaya’s work languished, no other techniques emerged for solving the basic problem of creating transversality and counting fixed points. Given the lack of tools they could trust and understand, most symplectic geometers retreated from this area, focusing on the limited class of problems they could address without recourse to Fukaya’s work. For individual mathematicians building their careers, the tactic made sense, but the field suffered for it. Abouzaid describes the situation as a collective action problem.

“It’s completely reasonable for one person to do this, it’s completely reasonable for a small number of people to do that, but if you end up in a situation where 90 percent of the people are working in small generality from a small number of cases in order to avoid the technical things that are done by the 10 percent minority, then I’d say that’s not very good for the subject,” he said.

By the late 2000s, symplectic geometers had worked through most of the problems they could address independently of the foundational questions involved in Fukaya’s work.

“Usually people go for the low-hanging fruits, and then the fruits hang a little higher,” Hofer said. “At some point, a certain pressure builds up and people ask what happens in the general case. That discussion took a while, it sort of built up, then more people got interested in looking into the foundations.”

Then in 2012 a pair of mathematicians broke the silence on Fukaya’s work. They gave his proof a thorough examination and concluded that, while his general approach was correct, the 1996 paper contained important errors in the way Fukaya implemented Kuranishi structures.

A Break in the Field

In 2009 Dusa McDuff attended a lecture at the Mathematical Sciences Research Institute in Berkeley, California. The speaker was Katrin Wehrheim, who was an assistant professor at the Massachusetts Institute of Technology at the time. In her talk, Wehrheim challenged the symplectic geometry community to face up to errors in foundational techniques that had been developed more than a decade earlier. “She said these are incorrect things; what are you going to do about it?” recalled McDuff, who had been one of Wehrheim’s doctoral thesis examiners.

Stéphanie Vareilles - CIRM

Dusa McDuff, a mathematician at Barnard College, struggled for years to fix what she saw as gaps in the foundations of symplectic geometry.

For McDuff, the challenge was personal. In 1999 she’d written a survey article that had relied on problematic foundational techniques by another pair of mathematicians, Gang Liu and Gang Tian. Now, 10 years later, Wehrheim was pointing out that McDuff’s paper — like a number of early papers in symplectic geometry, including Fukaya’s — contained errors, particularly about how to move from local to global counts of fixed points. After hearing Wehrheim’s talk, McDuff decided she’d try to correct any mistakes.

“I had a bad conscience about what I’d written because I knew somehow it was not completely right,” she said. “I make mistakes, I understand people make mistakes, but if I do make a mistake, I try to correct it if I can and say it’s wrong if I can’t.”

McDuff and Wehrheim began work on a series of papers that pointed out and fixed what they described as mistakes in Fukaya’s handling of transversality. In 2012 McDuff and Wehrheim contacted Fukaya with their concerns. After 16 years in which the mathematical community had ignored his work, he was glad they were interested.

“It was around that time a group of people started to question the rigor of our work rather than ignoring it,” he wrote in an email. “In 2012 we got explicit objection from K. Wehrheim. We were very happy to get it since it was the first serious mathematical reaction we got to our work.”

To discuss the objections, the mathematicians formed a Google group in early 2012 that included McDuff, Wehrheim, Fukaya and Ono, as well as two of Fukaya’s more recent collaborators, Yong-Geun Oh and Hiroshi Ohta. The discussion generally followed this form: Wehrheim and McDuff would raise questions about Fukaya’s work. Fukaya and his collaborators would then write long, detailed answers.

Whether those answers were satisfying depended on who was reading them. From Fukaya’s perspective, his work on Kuranishi structures was complete and correct from the start. “In a math paper you cannot write everything, and in my opinion this 1996 paper contained a usual amount of detail. I don’t think there was anything missing,” he said.

Others disagree. After the Google group discussion concluded, Fukaya and his collaborators posted several papers on Kuranishi structures that together ran to more than 400 pages. Hofer thinks the length of Fukaya’s replies is evidence that McDuff and Wehrheim’s prodding was necessary.

“Overall, [Fukaya’s approach] worked, but it needed much more explanation than was originally given,” he said. “I think the original paper of Fukaya and Ono was a little more than 100 pages, and as a result of this discussion on the Google group they produced a 270-page manuscript and there were a few hundred pages produced explaining the original results. So there was definitely a need for the explanation.”

Abouzaid agrees that there was a mistake in Fukaya’s original work. “It is a paper that claimed to resolve a long-standing problem, and it’s a paper in which this error is a gap in the definition,” he said. At the same time, he thinks Kuranishi structures are, generally speaking, the right way to deal with transversality issues. He sees the errors in the 1996 paper as having occurred because the symplectic geometry community wasn’t developed enough at the time to properly review new work.

“The paper should have been refereed much more carefully. My opinion is that with two or three rounds of good referee reports that paper would have been impeccable and there would have been no problem whatsoever,” Abouzaid said.

In August 2012, following the Google group discussion, McDuff and Wehrheim posted an article they’d begun to write before the discussion that detailed ways to fix Fukaya’s approach. They later refined and published that paper, along with two others, and plan to write a fourth paper on the subject. In September 2012, Fukaya and his co-authors posted some of their own responses to the issues McDuff and Wehrheim had raised. In Fukaya’s mind, McDuff and Wehrheim’s papers did not significantly move the field forward.

“It is my opinion that the papers they wrote do not contain new and significant ideas. There is of course some difference from earlier papers of us and other people. However, the difference is only on a minor technicality,” Fukaya said in an email.

Hofer thinks that this interpretation sells McDuff and Wehrheim’s contributions short. As he sees it, the pair did more than just fix small technical details in Fukaya’s work — they resolved higher-level problems with Fukaya’s approach.

“They understood very well the different pieces and how they worked together, so you couldn’t just say: ‘Here, if that’s problematic, I fixed it locally,’” he said. “You could also know then more or less where a possible other problem would arise. They understood it on an extremely high level.”

The difference in how mathematicians evaluate the significance of the errors in Fukaya’s 1996 paper and the contributions Wehrheim and McDuff made in fixing them reflects a dichotomy in ways of thinking about the practice of mathematics.

“There are two conceptions of mathematics,” Abouzaid said. “There’s mathematics as: The currency of mathematics is ideas. And there’s mathematics as: The currency of mathematics is proofs. It’s hard for me to say on which side people stand. My personal attitude is: The most important thing in mathematics is ideas, and the proofs are there to make sure the ideas don’t go astray.”

Fukaya is a geometer with an instinct to think in broad strokes. Wehrheim, by contrast, is trained in analysis, a field known for its rigorous attention to technical detail. In a profile for the MIT website Women in Mathematics, she lamented that in mathematics, “we don’t write good papers anymore,” and likened mathematicians who doesn’t spell out the details of their work to climbers who reach the top of a mountain without leaving hooks along the way. “Someone with less training will have no way of following it without having to find the route for themselves,” she said.

These different expectations for what counts as an adequate amount of detail in a proof created a lot of tension in the symplectic geometry community around McDuff and Wehrheim’s objections. Abouzaid argues that it’s important to be tactful when pointing out mistakes in another mathematician’s work, and in this case Wehrheim might not have been diplomatic enough. “If you present it as: ‘Everything that has appeared before us is wrong and now we will give the correct answer,’ that’s likely to trigger some kinds of issues of claims of priority,” he said.

Wehrheim declined multiple requests to be interviewed for this article, saying she wanted to “avoid further politicization of the topic.” However, McDuff thinks that she and Wehrheim had no choice but to be forceful in pointing out errors in Fukaya’s work: It was the only way to get the field’s attention.

“It’s sort of like being a whistleblower,” she said. “If you point [errors] out correctly and politely, people need not pay attention, but if you point them out and just say, ‘It’s wrong,’ then people get upset with you rather than with the people who might be wrong.”

Regardless of who gets credit for fixing the issues with Fukaya’s paper, they have been fixed. Over the last few years, the dispute surrounding his work has settled down, at least as a matter of mathematics.

“I would say it was a somewhat healthy process. These problems were realized and eventually fixed,” Eliashberg said. “Maybe this unnecessarily caused too many passions on some sides, but overall I think everything was handled and things will go on.”

New Approaches

A developing field does not have many standard results that everyone understands. This means each new result has to be built from the ground up. When Hofer thinks about what characterizes a mature field of mathematics, he thinks about brevity — the ability to write an easily understood proof that takes up a small amount of space. He doesn’t think symplectic geometry is there yet.

“The fact is still true that if you write a paper today in symplectic geometry and give all the details, it can very well be that you have to write several hundred pages,” he said.

For the last 15 years Hofer has been working on an approach called polyfolds, a general framework that can be used as an alternative to Kuranishi structures to address transversality issues. The work is nearing completion, and Hofer explains that his intention is to break symplectic geometry into modular pieces, so that it’s easier for mathematicians to identify which pieces of knowledge they can rely on in their own work, and easier for the field as a whole to evaluate the correctness of new research.

“Ideally it’s like a Lego piece. It has a certain function and you can plug it together with other things,” he said.

Polyfolds are one of three approaches to the foundational issues that have vexed symplectic geometry since the 1990s. The second is the Kuranishi structures, and the third was produced by John Pardon, a young mathematician at Princeton University who has developed a technique based on Wehrheim and McDuff’s work, but written more in the language of advanced algebra. All three approaches do the same kind of thing — count fixed points — but one approach might be better suited to solving a particular problem than another, depending on the mathematical situation.

In Abouzaid’s opinion, the multiple approaches are a sign of the strength of the field. “We are moving away from these questions of what’s wrong, because we’ve gotten to the point where we have different ways of approaching the same question,” he said. He adds that Pardon’s work in particular is succinct and clear, resulting in a tool that’s easy for others to wield in their own research. “It would have been fantastic if he’d done this 10 years before,” he said.

Abouzaid thinks symplectic geometry is doing well along other measures as well: New graduate students are coming into the field, senior researchers are staying, and there’s a steady stream of new ideas. (Though Fukaya, after his experiences, holds a different view: “It is hard for me to recommend my students go to that area because it’s dangerous,” he said.)

For Eliashberg, the main attraction of symplectic geometry remains, in a sense, the uncertainty in the field. In many other areas of mathematics, he says, there is often a consensus about whether particular conjectures are true or not, and it’s just a question of proving them. In symplectic geometry, however, there’s less in the way of conventional wisdom, which invites contention, but also creates exciting possibilities.

“For me personally, what was exciting in symplectic geometry is that whatever problem you look at, it’s completely unclear from the beginning what would be the answer,” he said. “It could be one answer or completely the opposite.”

Update and correction: On February 10 this article was updated to include the work of Andreas Floer and to clarify the timing of the various papers that were posted following the 2012 Google group discussions.

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  • Great article! Wish you had mentioned Floer's name in there somewhere, maybe with Gromov's. Thanks for sharing this story.

  • I've read the article, it's nicely written and contains nontrivial stuff, much of which was new for me as a trained theoretical physicist. But sorry to say, I don't think that this text contains persuasive evidence that would make me agree that it has justified the ambitious claims in the title and elsewhere.

    To put it concisely, I find it likely that I would stand squarely on Fukaya's side even if I understood all these things in detail.

    Symplectic geometry as started by Hamilton is really simple in principle, intuitive part of mathematics or geometry that may be dealt with as a subset of the science with manifolds, bundles or functions on them, and analyses of the functions. At least for differentiable finite-dimensional spaces, all the questions and quantities, like the intersection points, seem well-defined, and the general strategies to resolve the singular situations, i.e. with locally coincident contour, seems self-evidently correct. Even if it's hard to find the answers in classes of examples, it wouldn't justify the statement that something is wrong with the foundations. It's often true that it's hard to find answers to questions. But the foundations don't include methods to answer all questions, this would be too much to ask.

    This symplectic geometry field was taken by mathematicians but it doesn't mean that the direction of progress must involve the excessive, Bourbaki-style focus on rigorous formulations of everything. I think that Gromov but also Fukaya have actually advanced the field in a way that may be appreciated even by the physicists – and would be appreciated by Hamilton – and not just in some bureaucratically rigorous way of rewriting everything.

    Some lady or ladies arrives (I will emphasize that the gender should be irrelevant) and starts to yell that things aren't rigorous enough for her. Maybe someone finds a mistake in a paper with lots of pages – which however changes nothing about the major ideas and their cleverness. Sorry, I don't think that this kind of opposition is valuable. One may always arrive to any field and start to yell that things aren't great or rigorous enough. Is it helping things? Is it a contribution to science or mathematics? And is the critic actually more rigorous than the criticized person, or does the critic just assume that by being a critic, she will be assumed to be the better one?

    I also utterly disagree with the claim that because Fukaya had to release – and was kind and patient to release – hundreds of pages of detailed explanations means that his previous work wasn't brilliant or at least morally correct. Some listeners just need far more detailed explanations. But that's often their fault. Over the years, I have written the equivalent of tens of thousands of pages of explanations of things that – I believe – truly intelligent people get in minutes and without much help. The size doesn't mean that something is wrong about the explained issues.

    And why is the article referring to "women in mathematics"? I am sorry but this is quite some circumstantial evidence to figure out where the wind is blowing from. If some people in the system want to make a mathematician – possibly an unconstructive if not obnoxious mathematician – look more visible just because she is female, it is wrong, wrong, wrong.

    Again, the article hasn't provided me with persuasive evidence that the people who ignored these complaints were wrong. I tend to think that they were – and they still are, I guess – right and there aren't really any big gaps in the foundations of symplectic geometry.

  • Thanks @adam saltz. We have added a mention of Floer's work, along with a note at the bottom alerting readers to the addition.

  • Luboš,

    Your comment is misguided. First of all, this is an article about mathematics, not physics, so rigor is indeed important (and they do not write in the Bourbaki style anyway). Second, it is clear that "closed orbit of a Hamiltonian flow" is well defined, and the foundations of such are fine. The issue is that there was complicated machinery invented to address this question (and many others), and the foundation in this case is the proof the machinery itself actually makes sense. This is not too much to ask.

    And just a little outside perspective: You seem strangely fixated on the fact that McDuff and Wehrheim are women. Believe me, they are entirely prominent without expository articles such as this.

  • Jonathan, Dusa McDuff is a senior figure in that field but just open scholar dot google dot com and search for katrin-wehrheim and kenji-fukaya, respectively. Wehrheim is not really in the same league as Fukaya.

    I didn't start with the women theme. The very article above referred to "Women In Mathematics" which is a feminist tabloid selectively trying to emphasize women, as the name makes obvious, regardless of the research merits. You may think that it's just OK but serious scholars don't find it acceptable and it's not acceptable for the Quanta Magazine to rely upon similar dubious sources, either. Mathematics and science shouldn't be contaminated by these ideological issues and biases.

    Again, rigor as understood by narrow-minded mathematicians is not a necessary condition in this kind of mathematics and the best researchers mostly agree with me, not with you. These questions have been discussed ad nauseum. For a major example, Google search for

    "Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics"

    and especially

    "Responses to Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics, by A. Jaffe and F. Quinn"

    where many authors give their opinions. Most of them actually endorse even speculative work. But this is not really what's being debated here. The question is whether one allows methods that are on par with heuristic methods used by theoretical physicists, but otherwise very systematically and carefully applied ones. The answer is obviously Yes, those are an important branch of the research.

    Fukaya's methods have very good reasons to be followed in roughly thousands of articles and it's just wrong and it should be ignored if someone would want to delegitimize this whole branch of research.

  • This article is very interesting and as all good writing, it inspires new thought.
    The following came to my mind.

    1) Why was the work of Fukaya ignored for 16 years? This brings to mind the more recent work of Mochizuki, which also is ignored in the west. One might speculate, that, since the best mathematicians in the West go to work for threeletter agencies, there are just a few left, who just don't have the capacity to solve the problem.

    2) Why exactly is Fukaya saying that symplectic geometry is "too dangerous"? It would be nice to read more about the reasons for calling symplectic geometry "too dangerous".

    3) There is a difference between an author writing a book and the people doing the proof reading. The same rules for authorship should apply in the Mathematics world. Making spelling mistakes, should not mean that the proofreader can steal the authorship from the author. This lack of standards is something Grothendieck complained about, made Perelman drop out of Mathematics, and might also be the reason why Fukaya labels a career in symplectic geometry as "too dangerous". Or, using the metaphor, from the article, who goes into history: the one, who first conquers the mountain or the one who leaves the hooks along the way? There definitely seem to exist a lack of standards, which might allow academic theft in the mathematics community.

    4) Like it or not, there seem to be a cultural difference between Japanese and western mathematics. And a big part of the problem seems to be jealosy and pride, which in part is dependent on culture. It also seems that Japanese mathematicians do not accept that people are being unpolite to them.

    5) Now, imagine a young talented student, our future Grothendieck. He of course sees, that coming up with new bold and groundbreakin maths, might be bad for his career and wellbeing, that his work will be ignored, and in the end the proofreader might steal your proof. He will see what has happened for Grothendieck, Perelman, Fukaya and Mochizuki. I believe that our future Grothendieck will think "who would want to do maths in such a toxic community". He will drop his math ambitions and go working for the private sector as the math career is "too dangerous", borrowing the words of Fukaya.

    Now this is actually a very real problem for the maths community, which is not addressed. Mochizuki and also some of his fellow colleagues from Kyoto has actually been writing about this problem. However I do believe QM could make a difference.

    Fininshing this peace off, it would be great, if QM would write more about
    1) envy, authorship, intellectual theft and (the lack of) ethics in the Mathematics community,
    2) why many of the most talented mathematicians will rather leave university research, than risking that their career will be ruined by publishing groundbreaking mathematics as so many times before.
    3) the slow-down effect of three-letter agencies hiring the most talented mathematicians (who didn't leave mathemathics) on the quality of the maths, which is not conducted in secret and published in peer-reviewed journals.
    4) cultural differences between the West and Japan (and other countries too) and the importance of being polite etc.
    5} the "impact ratings" as the career engine and the accompanying discouragement of trying out new cross-diciplinary ideas. Mochizuki has written about this.

    Those were my thoughts. Thanks again fot this thought-provoking piece.

  • Interesting article by Kevin Hartnett, interesting comments by many including Lubos and Jonathan. My own enchantment is about the progress that has been made in all fields of human endeavor and the dedication of the participants through history. Very commendable indeed.

  • So the author's conclusions were correct, and his original technique is effective and will be used by others to solve similar problems.

    I'm not sure if this article has anything to add!

  • The article describes things quite cautiously, yet without the comment from mr. Motl I would have been quite misguided. I am no longer sure, that the clarification really is or will be as important as it seemed on first glance. We may see it in future, if it was really the " breakthrough" ( at least in understanding and acceptance), as it is maintained to be.

  • I appear to be the first person commenting on this article who is actually a tenured mathematician doing research in symplectic geometry. I've been following this whole story for several years while not getting involved in it myself; I know all of the people involved, including everyone who is quoted or mentioned in the article, and I know something about the mathematical issues. Before saying what I want to say, however, let me preempt Luboš Motl's inevitable appeal to "scholar dot google dot com" and freely acknowledge that I am not in the same league as Fukaya. Nor am I, for that matter, in the same league as Wehrheim. (But at least I have no association with any website called "Women in Mathematics", so I guess Luboš would say I have that going for me?) (???)

    But this is not just a response to Luboš; I have a more general remark for several people who have written comments above about the general value of groundbreaking but imprecise mathematical ideas, and how it is the reader's fault when they cannot understand things which would be clear to a "truly intelligent" person, and so forth. Actually, I have two questions.

    Do any of you actually have any understanding of the technical issues involved in the proof of the Arnold conjecture?

    If not, then why do you feel entitled to express an opinion on the value of the discussion that McDuff and Wehrheim started?

    Seriously, that takes a lot of nerve. Luboš demonstrated very clearly in his first comment that he knows little or nothing about Floer homology, how to prove that moduli spaces of pseudoholomorphic curves are smooth, why the standard arguments fail in the presence of multiply covered curves, what kinds of methods people have proposed for dealing with that, etc… I do not mean it at all as an indictment of his character when I observe that he isn't aware of these things, but he ought at least to be aware of it when his knowledge is insufficient for drawing a conclusion. Instead, he read Kevin Hartnett's attempt to boil complicated transversality issues down into simpler ideas that a general audience might grasp, and assumed he had therefore understood the main issues, which McDuff and Wehrheim obviously also should have understood if they weren't being so nitpicky. What a mindbogglingly arrogant thing to think. But this is the internet, of course, so actually understanding a complex topic is obviously not considered a prerequisite for putting your Opinion out there and putting someone down in the process.

    I would respectfully request from the lot of you that before telling us your generalized opinions on how not to stifle the creativity of the young Grothendiecks among us, you should learn enough about the actual subject at hand to judge whether these generalized opinions are in any way relevant. And then, be prepared to deal with followup questions such as, "but what should a PhD student in symplectic geometry do when he/she cannot understand the crucial paper in which Fukaya-Ono proved the Arnold conjecture for general symplectic manifolds, and none of his/her mentors understand it well enough to explain it?"

    There's one more thing I would like to clarify: several people seem to have gotten an inaccurate impression from what this article says about issues of "who should be given credit". To my knowledge, McDuff and Wehrheim have never claimed to be seeking credit for anything like proving the Arnold conjecture. Their primary goal the entire time seems to have been to achieve a wider understanding of the important work that was done — and to fix it wherever fixing is necessary — because these technical issues have potential implications far beyond the solution to this one particular problem, and it would be important for the symplectic community at large to understand them. Their stated motivation in writing papers about it and getting them published has always been so that the ideas they are developing and refining can be checked independently via the peer review process, so that in the long run, we can develop a widespread consensus about what is correct. And to be clear, these are not "minor" technical issues: the major breakthrough in solving the Arnold conjecture was made by Floer, several years before Fukaya and Ono. The point of the latter's paper was supposed to be to explain how you solve the TECHNICAL problem of removing certain unwanted assumptions from Floer's arguments. That technical problem turns out to be quite hard, evidently harder than Fukaya and Ono believed it was at the time, and I think it is fair to say that if the vast majority of the community cannot understand their arguments on close inspection, then the arguments are not yet complete. I don't say this out of any desire to take credit away from Fukaya and Ono — who certainly have contributed invaluable ideas to the field both in that and in other papers — but I do think that if later mathematicians spend years of their careers trying to fill the gaps that the original version left unfilled, and they eventually succeed in making the techniques accessible and usable for the wider symplectic community, then that is a valuable effort that should be rewarded. If the mathematical community chooses not to value that kind of work, then that is a major character flaw on the part of the community.

  • As another mathematician working in a somewhat nearby field (let's say, geometric topology) I am thankful to Chris Wendl for rescuing this thread from the rantings of know nothings.

  • Dear Dr Wendl, Google Scholar is a top comprehensive source of information about the scientific literature and its influence. Indeed, it also shows that Fukaya's contributions are so much above yours that you don't really have much mathematical capital to question his work, either. Whether it was right to hire you or somebody else is an issue that you, and not I, have raised but be sure that many others would have doubt.

    Equally importantly, one may extract pretty much the same picture from other databases of mathematical papers and citations. Your attempt to dismiss this argument because Google Scholar isn't good enough for you is simply bogus.

    I am not an expert in the Arnold Conjecture but I am still more or less sure to know significantly more about these technical topics than the author of the text, K.H. (shall we make a contest involving some questions and answers about symplectic geometry or Morse theory etc.?), and if he feels entitled to question the value and robustness of the work by Dr Fukaya, don't be surprised that I fell to be at least equally entitled to question the value of his and the two critics' criticisms of that program. I don't question that technical questions in this enterprise are hard. But things' being hard is very different from saying that some criticism is valuable. The article has not really presented any evidence that the criticism is valuable.

    As some other scholars pointed out on my blog, this kind of "looking for mistakes and fixing them" was the main thing that drove Perelman out of the Academia, too. But this culture was limited when he was active relatively to this new "culture".

    Things are valuable to a different extent but the claim – implicitly or explicitly included in this article or your comment – that the two critics' efforts to fix the mistakes have the same or larger values than the contributions by Fukaya and Floer and others is simply indefensible. Google Scholar or any equivalent source *is* a legitimate tool to discuss the relative importance of research when the differences are this large. And when a journalist tries to place the efforts to "fix" things above the original contributions given these conditions, he or she is simply deceiving his or her readers.

    And I've spent a long enough time in the Academia to be pretty much sure why the journalist found it right to distort the situation in this way.

  • As another tenured mathematician in a nearby field, I'd also like to thank Chris for his comment. Jonathan too. (And Adam for his enthusiasm.)

    Going forward, I think maybe it's best to just ignore Lubos, who is taking a position on a mathematical question despite having no understanding of the relevant mathematics (unlike Kevin Hartnett, who is a science writer reporting on the sociological fact that there is disagreement in the symplectic geometry as to the significance of the gaps in Fukaya's work; this is apparently a distinction Lubos fails to understand). Every post of Lubos's contains a surprising number of logical reasoning and reading comprehension errors (surprising given that he used to be a string theorist and therefore presumably somewhat skilled in logic, right?) that are obvious even to people with no mathematical or scientific background. Is there really a pressing need to refute them here?

    As to the article itself: I thought it was great. An impressive attempt to convey in a simple manner complicated mathematical issues that are deep in the weeds of symplectic geometry, as well as a balanced look at a controversy which points to a more general philosophically difficult question: what constitutes proof in mathematics? Most impressive perhaps: none of the actors in the controversy come off as looking bad in this article! Fukaya's work has been very important; McDuff and Wehrheim's work to close gaps in the work and make it understandable to a wider audience of symplectic geometers is also a valuable contribution to mathematics.

  • As another tenured mathematician in a nearby field, I'd also like to thank Chris for his comment. Jonathan too. (And Adam for his enthusiasm.)

    Going forward, I think maybe it's best to just ignore Lubos, who is taking a position on a mathematical question despite having no understanding of the relevant mathematics (unlike Kevin Hartnett, who is a science writer reporting on the sociological fact that there is disagreement in the symplectic geometry community as to the significance of the gaps in Fukaya's work; this is apparently a distinction Lubos fails to understand). Every post of Lubos's contains a surprising number of logical reasoning and reading comprehension errors that are obvious even to people with no mathematical or scientific background. Is there really a pressing need to continue to refute them here?

    As to the article itself: I thought it was great. An impressive attempt to convey in a simple manner complicated mathematical issues that are deep in the weeds of symplectic geometry, as well as a balanced look at a controversy which points to a more general philosophically difficult question: what constitutes proof in mathematics? Most impressive perhaps: none of the actors in the controversy come off as looking bad in this article! Fukaya's work has been very important; McDuff and Wehrheim's efforts to close gaps in Fukaya's work and make it understandable to a wider audience of symplectic geometers is also a valuable contribution to mathematics.

  • Regarding the issue of the perturbed function in the above diagram to achieve a finite intersection; wouldn't a measurement that uses a Plank length as a division unit give a finite result?

  • Wow… I had assumed that being directly invited to unleash an ad hominem attack against me would discourage Luboš from actually doing so, but the result is more hilarious than I expected. John Baldwin is of course correct in theory that the proper response to Luboš's rants is to let him embarrass himself and otherwise ignore him, but since one or two other people on this thread seem to have appreciated what he has to say, I feel compelled to respond to one detail in his last comment:

    "the claim – implicitly or explicitly included in this article or your comment – that the two critics' efforts to fix the mistakes have the same or larger values than the contributions by Fukaya and Floer and others is simply indefensible"

    That is a claim that no one has made. No one in the world, ever. Not McDuff, not Wehrheim, nor anyone in the article, nor me, neither explicitly nor implicitly. This is a straw man argument, a deliberate attempt to tar the reputation of two very well-respected mathematicians, and utterly disconnected from reality. I'm a little disappointed in the moderators of this comment section for permitting it.

    I also have to say that while my initial reaction to the article matched John's pretty well, this comment thread has been gradually making clear to me some flaws that weren't apparent at first. There are a lot of inaccuracies (e.g. the conclusion that the problems with the Fukaya-Ono article have meanwhile been fixed is by no means unaminous among symplectic geometers), and things that are stated vaguely enough to give a potentially harmful false impression (e.g. the issue about credit). What I've seen in these comments is people running away with those false impressions to justify forming a lynch mob, and it is alarming.

  • For the record: Chris Wendl asked, suposedly me, the following "but what should a PhD student in symplectic geometry do when he/she cannot understand the crucial paper in which Fukaya-Ono proved the Arnold conjecture for general symplectic manifolds, and none of his/her mentors understand it well enough to explain it?"

    Well I'd say in that case I'd contact the author of the crucial paper and ask for some clarification of the difficult parts. In the aritcle we discuss, we read that Fukaya waited 16 years for someone to call. Wendl's question goes some way towards explaining why it really took those 16 years for someone to get that mingbogglingly groundbreaking idea of picking up the phone.

  • Anonymous Reader,

    Your previous comment demonstrated quite a few unwarranted assumptions based on insufficient information, though it is not necessarily your fault in every case. My information on this detail is admittedly second-hand, but one of the subtly inaccurate impressions conveyed by this article is the notion that no one in those intervening 16 years attempted to contact Fukaya for clarification of his work. Apparently, people did. I have no information on whether any of them got a response, and if so whether that response was helpful. But I do know that Fukaya et al never began producing clarifications that could be of any potential value to the wider community until McDuff and Wehrheim raised their concerns, and we are now better off for it.

  • @A reader

    > 5} the "impact ratings" as the career engine and the accompanying discouragement of trying out new cross-diciplinary ideas. Mochizuki has written about this.

    Could you give a the concrete reference on this, since I would love to read what concretely Mochizuki wrote that topic.

  • as someone who left mathematics with a ph.d and firmly in the camp of proof over idea as referenced in the article, I have stumbled on many articles (reviewed, published, celebrated?) that fall in the idea category.

    i privately termed that "the russian school" since it seems to be a tradition favored there as well.

    my experience is that mathematicians in the idea camp spend a lot of time talking to other mathematicians convincing them of the soundness and correctness of their ideas. and if that's generally true, doesn't that just mean that you can only truly understand their theorems by having direct access to them?

    so why preclude the rest of the community from understanding what is going on?
    conspiratorally [tm] it seems like the idea-mathematician want the credit but do not want to truly spread the knowledge…

    but is the problem with the writing down a proof, that it is hard? isn't that what ph.d. students are for? clearing the mines to the field it isn't dangerous any more, if you will.
    or are the ideas not really, truly good ones if you can't prove them in less than 256 pages? where's the beauty in that, if you will.

    i seriously think it is a huge problem for the community that it is fractured into groups (cliques?) that aren't on the same page when it comes to rigor of proof.

  • Lubos Motl

    It seems that you are against mathematical rigor in Sympletic Geometry. Do you argue that comming up with clear precise definitions at the foundations of Symplectic Geometry is a waste of time? Did not the rigorous foundations of Witten's heuristic approach to three manifolds and knots provided by Michael Atiyah launch a thousand ships of deep mathematical understanding?

    Can you define a Fukaya Category to Dusa Mcduff during a phd oral comprehensive exam?

  • While I have had entanglements with Lubos in the past that doesn't mean his points are without merit. Often he takes a strident but generally truthful attitude towards things. So here I will add my layman forceful opinion. And I frankly couldn't care less whether or not a mathematician thinks I should or should not offer it. What stands glaringly in your face in this article is that a man who produced brilliant work more than a decade ago without anyone bothering to contact him about it, but then be faced with the prospect of a newcomer that is simply trying to understand his work questioning and defining just how much credit he truly deserves. Just becomes these new authors have proven more adept at communicating with a western audience in no way detracts his contribution to the field. This has been stated before but it is so glaringly obvious that it bears repeating. If someone did not fully understand the proof why not contact the original author and collaborate on a clarification, all the while admitting the truth that the original already contained the critical breakthrough. Is that really too much to ask?

  • As I've already indicated in a previous comment, the notion that no one attempted to grapple with Fukaya's ideas in that intervening period or to ask him about them is simply inaccurate. Among other things, Fukaya and his collaborators gave talks at many conferences all over the world during that time, and interacted heavily with other symplectic geometers, as they still do. Moreover, Wehrheim and McDuff(!) were not "newcomers" when they started this project. It's rather important in fact that they weren't, as anyone more junior raising such questions would have been as easy for the symplectic community to dismiss as it apparently is for the lay readers of this comment thread.

    Ignacio Mosqueira: I respect everyone's right to their own opinion, mathematician or otherwise, but it is reasonable for me to point out when that opinion is based on unwarranted assumptions in the presence of insufficient knowledge.

  • Chris Wendl. Did you personally invite the author to clarify these points to you? Or are you giving the newcomers to the proof credit for knowing what to ask? Once again if you yourself did not understand the proof but you thought it was important to do so then why didn't you fire off and email to the author? Or is he responsible for the laziness of others? Look there is absolutely nothing that you can tell me that will convince me that the person mostly responsible for a breakthrough in math is somehow to blame because it took others a long time to appreciate his insights. This claim is warped in the face of it. It strikes as a truly irksome claim to level at a person who has made a significant contribution to your own field.

  • @Wolfgang Keller, here is the requested quotation by Mochizuki about why there is so little progress in Maths, from his paper "The mathematics of mutually alien copies" (http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf), page 112:

    Escaping from the cage of deterministic models of mathematical development: The adoption of strictly linear evolutionary models of progress in mathematics […] tends to be highly attractive to many mathematicians in light of the intoxicating simplicity of such strictly linear evolutionary models, by comparison to the more complicated point of view […]. This intoxicating simplicity also makes such strictly linear evolutionary models — together with strictly linear numerical evaluation devices such as the “number of papers published”, the “number of citations of published papers”, or other likeminded narrowly defined data formats that have been concocted for measuring progress in mathematics — highly enticing to administrators who are charged with the tasks of evaluating, hiring, or promoting mathematicians.

    Moreover, this state of affairs that regulates the collection of individuals who are granted the license and resources necessary to actively engage in mathematical research tends to have the effect, over the long term, of stifling efforts by young researchers to conduct long-term mathematical research in directions that substantially diverge from the strictly linear evolutionary models that have been adopted, thus making it exceedingly difficult for new “unanticipated” evolutionary branches in the development of mathematics to sprout.

    Put another way, inappropriately narrowly defined strictly linear evolutionary models of progress in mathematics exhibit a strong and unfortunate tendency in the profession of mathematics as it is currently practiced to become something of a self-fulfilling prophecy — a “prophecy” that is often zealously rationalized by dubious bouts of circular reasoning.

    In particular, the issue of escaping from the cage of such narrowly defined deterministic models of mathematical development stands out as an issue of crucial strategic importance from the point of view of charting a sound, sustainable course in the future development of the field of mathematics, i.e., a course that cherishes the priviledge to foster genuinely novel and unforeseen evolutionary branches in its development.

    It is advisable to read the full paragraph §4.4 on pages 109-112 in Mochizuki's paper to better understand his reasoning.

  • Ignacio

    Dusa Mcduff is a giant in the field of Symplectic Geometry…also married to a well known Fields Medalist. Werhiem…a tenured MIT math professor…was one her phd students.

  • And again: the person "mostly responsible" for the breakthrough in solving the Arnold conjecture was not any of the people we are discussing here. It was Andreas Floer, after whom the very large subject of "Floer homology" is very rightly named.

    Floer's proof (around 1989) introduced revolutionary new ideas that reshaped the field of symplectic topology, but they were not originally applicable in as much generality as one would like. Gromov's related theory of pseudoholomorphic curves, which had already revolutionized the field a few years before Floer, had the same drawback. Removing these restrictions was a difficult technical problem that several people attacked and published papers about during the 1990's, not only Fukaya and Ono. As of today, there is no widespread consensus in the symplectic community as to which of those approaches was best or whether any of them is free of major gaps. It is universally acknowledged that all of them contain minor errors, but of course that is not the main concern. When minor errors in a paper are widely recognized as being fixable, no one raises a fuss about it, and honestly, it's insulting that so many people from the outside just assume that's what's going on here. When crucial definitions and important arguments are frequently stated too vaguely for the reader to be sure what the author means, much less to be able to check their correctness, many readers simply get discouraged and decide to spend their time thinking about something else. That is why a consensus failed to develop. In a vacuum, it might be valid to say that anyone who truly cares about the mathematics should simply ask the author for clarification… but there comes a time when it starts to seem like more effort for the reader to understand the paper than it was for the author to write it, and that should not be. Those readers have other things to do, their time is valuable, they need to publish papers of their own, or else they won't get jobs.

    I'm explaining this because it's apparent that for people who do not know the subject and have not been hearing about this controversy already for years, the article paints a severely oversimplified picture.

  • Patrick it makes absolutely no difference for the purposes of this article where each of the authors works. It's irrelevant information. The contribution of each author does not depend of the institution they hail from. If McDuff and her student had made other contributions to the field that that's great for them. Here they simply helped to clarify the pioneering work of someone else.

    Chris Wendl thanks for the explanations they are instructive and interesting. They do not change the substance of the argument in any way. I doubt very much that anyone is trying to undercut the contribution of Floer. Certainly there was no such claim made in the article I am commenting on.

  • Wow.. I really think the mathematicians here might save their energy and return to more useful pursuits, I do not think it will be possible to make this angered mob see reason, and the unpleasantness and in parts lowness of the whole thing gives me hives when reading (hey moderated comment section, where is your moderator?)…all arguments people are making in good faith to help clarify the situation being twisted around…might as well try to empty the Mediterranian with a spoon.

    because the comments section is like:
    mathematical community: hey author of article, you actually forgot to mention this pretty major contribution by Floer..also Gromov. Maaaaybe fix that! Love, your mathematical community.
    author of article: Hey, whassup, oopsies, let me stick the name Floer in there somewhere, also Gromov. Phew, really saved it with this afterthought, so that nobody will think my research wasn't thorough.
    reader 1: Hey, it's a battle, time to take sides and start flinging poop! There are those poor mathematicians slaving away for years, producing brill ideas, and then some lady mathematicians have the nerve to waltz in and nitpick their stuff, and then they want all the glory. How is that fair? Boo on the lady mathematicians!
    mathematical community: Hey, it was a bit different…the problems that McDuff and Wehrheim pointed out actually needed to be fixed, and look at them, they are fine mathematicians in their own right….
    reader 1: Nah, I googled them, maybe Senior Lady is fine, but Junior mathematical lady…pathetic! Boo on the lady mathematicians and their feminist platform!
    reader 2: Yeah, look at the whole mathematical community, pathetic! First never asking questions and then poo-poohing! Those ladies want to be glorified for doing proofreading, is what it is! Making the environment toxic, is what it is! Nipping future Grothendiecks in the bud, is what it is!
    mathematical community: Hey, we're mathematicians, so let us point out that maybe you are not quite informed about the toughness of the work that was involved in fixing those technical things, because hard technical stuff = mathematician's bread and butter, so maybe you should think twice before dismissing it as "proofreading". Also, who said McDuff and Wehrheim sought credit for proving the Arnold conjecture? They did not, you know? But, you know what would have been nice? Mentioning how Floer had actually the initial bright idea.
    reader 1: I googled you, you're pathetic. Also, if the author of this article can critique Fukaya, then I can totally tear into McDuff's and Wehrheim's work, laypersons both of us. Also also, claims in article and by mathematical community that McDuff and Wehrheim's contribution more valuable than Floer and Fukaya contribution ist just so wrong…
    mathematical community: What did we just say? Nobody said that. Reader 1, you are a bundle of flailing incoherence.
    reader 2: yeah right pathetic! Anything anybody had to do was email, you would've gotten all the answers…but NOBODY DIYAD.
    mathematical community: No, it is not like that. Let me make this measured statement of how people tried to establish contact and ask questions over the years, based on what I know, only what I know and not making assumptions.
    reader 3: Totally pathetic, agree with reader 1, am also not a mathematician, they could have emailed, or or…instead nothing for 10 years, and then those bad junior mathematician westerners steal everything from this poor Asian person, when they could just have asked questions…
    mathematical community: No, it was more like for over 10 years everybody tried everything to understand….also: for the record: McDuff and Wehrheim not junior.
    reader 3: totally pathetic that nobody contacted them, nor did you…
    reader 2: Nipping the young Grothendiecks, in the bud! Believe it's an east and west thing! Let me throw some philosophical stuff about the eastern and western churches at you! Unrelated, yes, and yet explains everything going on here!
    mathematical community: Again, McDuff mathematical giant, Wehrheim no mathematical orphan either. Don't exactly need to go around claiming other people's ideas for themselves. Also, the read breakthrough w.r.t. solving the Arnold conjecture coming from Floer…
    reader 3: Just because they work at fancy-pants universities and IF they made other contributions to the field does not mean they can get credit for other people's ideas!…

    etc. etc. etc. will no doubt go on forever

  • Not that it matters much, but Patrick O'Rafferty made two factually inaccurate statements: Wehrheim did her PhD at ETH with Dietmar Salamon, she was not McDuff's PhD student. And she did previously work at MIT, but she is now at Berkeley.

    I agree with Ignacio that such details are not especially relevant, just as I would point out to Lubos that the comparison between my academic reputation and Fukaya's would not be considered relevant by any journal that asked me to referee the latter's paper (which I have not done, but I have occasionally had to referee papers by people of similar stature, and I point out mistakes when I find them). After he characterized the people raising questions about Fukaya's work however as "newcomers", it is VERY relevant to point out that McDuff is one of the founding figures in the field and the co-author (with Salamon) of two textbooks that every grad student in the field must read.

  • Dear Patrick,

    you asked me: Is the perfectly rigorous treatment of some convincing semi-heuristic or physics-like results a waste of time? You must know how I actually feel about those things, right? My answer is obviously: Mostly yes, I do think that the perfectly rigorous incarnations of semi-heuristic ideas represent a much less valuable way to spend the time of very smart brains than the search for the solid enough yet semi-heuristic or physics-like ideas in mathematics. In particular, I don't believe that any "fully rigorous" work is at the cutting edge of these disciplines how I see them.

    This answer can't be shocking to you because quite a long time ago, I chose theoretical physics as the more balanced attitude to these matters than rigorous mathematics (and some semi-heuristic mathematicians are very close to the spirit of theoretical physics). In my eyes, rigorous mathematics has always resembled the filling of tax returns, at least to some extent. I've never thought that this "bureaucratic" work was too much fun, I didn't understand why people wanted to do it voluntarily, and I've never believed that the people doing this work were the true intellectual elite. So yes, I do think that the mathematicians or physicists who find the rough shape of some clever mathematical structures or operations or proofs are doing more important work than those who transform them, usually with a huge delay, into rigorous results.

    Yes, despite my respect for Atiyah who is amazing, I do believe that the results by Witten you mentioned are more important than Atiyah's subsequent work on those and I am obviously much less familiar with the latter.

    But I wouldn't go too far to claim that the rigorous work is worthless. There are cases in which some really important mistakes resulting from a non-rigorous approach are uncovered, other cases when the search for the rigorous proof produces tools that may be used much more generally in other situations, and even if none of these two situations takes place, it's simply a natural and legitimate activity to formulate results truly exactly. At least, the maximized rigor is a natural discipline of sports. I have no doubt that Atiyah is a great man even though his way of thinking and desires obviously do differ from those of us who think as physicists.

    But what I observed in this story and others is that almost completely analogous situations are treated very differently and these double standards almost clearly have some political, ideological, or egotist drivers. When people around Yau were fixing some details in Perelman's proof of Poincaré's conjecture, media such as The New Yorker ran hit pieces such as one titled "The Manifold Destiny" against Yau (2006) – with no justification – which were full of hostile lies about Yau. Yau has never claimed to be on par with Perelman in the process of proving the Poincaré conjecture or other things. While appreciating Perelman a lot, he (and/or his junior collaborators) has helped this proof to be completed and he has done lots of important things for mathematics and the institutions doing research on it.

    Now, when the Arnold conjecture is discussed and when the "auditors" seem to be both 1) less important for the fixing of the holes (whose larger part was still fixed by Fukaya and OOO) and 2) more aggressive in trying to claim a big part of the credit, the media suddenly write exactly the opposite. The authors of the original ideas are painted as rather bad and the auditors are "heroes". It just "happens" that most of the "villains" in the media stories are Asian and all the women in the media stories are always "heroines".

    Sorry but it's obvious that there are double standards here and something non-meritocratic is heavily affecting the "framing". This conclusion may be reached by a comparison: it doesn't really depend on how much one values rigorous results and the technical work on details and small mistakes relatively to the big-picture pioneering work. Whatever is one's opinion about the importance of the totally rigorous work, he or she should be able to see the inconsistency with which the journalists in particular (but not only journalists) were approaching very analogous situations.

    Best wishes
    Luboš

  • "momentum — how fast it’s moving and in what direction"

    Just a small point – the author has apparently defined velocity not momentum. Momentum is then the product of velocity and mass (in classical mechanics). The Hamiltonian indeed then relates generalised position and momentum co-ordinates.

  • By the way, this was a very interesting article, more please. It is a pity that one commenter has taken it as the basis for a political/ideological rant and personalised attacks.

  • @Chris Wendl

    I agree with 'another reader'. Replying to people like Ignacio etc. is pointless. His second comment is just regular trolling.

    There is always a problem with pop science article that talk about people over and above the science (cf. the New Yorker piece around the proof of Thurston's geometrization), even with good ones like this. It enables some people to believe that they have some significant insight to offer by substituting their own experience of workplace politics into academia – which they then go on to imagine as a locus of the most vicious of Darwinian struggles. As someone who spent three years in industry before beginning my phd, I find this picture hilarious. I have now seen a few departments and while there is a fair amount of politics (we are humans), it is in comparison preciously mild, even naive to some extent.

  • In the article we have a very strange situation described, where "90 percent of the people are working in small generality from a small number of cases in order to avoid the technical things that are done by the 10 percent minority".

    How come we ended up with such an absurd situation? Is it just an anomaly or a symptom of a wider problem?

    Here is a letter of a Ph.D. student detailing why he was nipped "in the bud", to borrow the words from the comment of "another reader":https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/

    The text should be enjoyed in its full length, here is an excerpt:
    "I cannot help but get the impression that the majority of us are avoiding the real issues and pursuing minor, easy problems that we know can be solved and published. The result is a gigantic literature full of marginal/repetitive contributions. This, however, is not necessarily a bad thing if it’s a good CV that you’re after.
    (
    5) Academia: The Black Hole of Bandwagon Research
    Indeed, writing lots of papers of questionable value about a given popular topic seems to be a very good way to advance your academic career these days. The advantages are clear: there is no need to convince anyone that the topic is pertinent and you are very likely to be cited more since more people are likely to work on similar things. This will, in turn, raise your impact factor and will help to establish you as a credible researcher, regardless of whether your work is actually good/important or not. It also establishes a sort of stable network, where you pat other (equally opportunistic) researchers on the back while they pat away at yours.
    Unfortunately, not only does this lead to quantity over quality, but many researchers, having grown dependent on the bandwagon, then need to find ways to keep it alive even when the field begins to stagnate. The results are usually disastrous. Either the researchers begin to think up of creative but completely absurd extensions of their methods to applications for which they are not appropriate, or they attempt to suppress other researchers who propose more original alternatives (usually, they do both). This, in turn, discourages new researchers from pursuing original alternatives and encourages them to join the bandwagon, which, though founded on a good idea, has now stagnated and is maintained by nothing but the pure will of the community that has become dependent on it. It becomes a giant, money-wasting mess."

    One of the commenters sums up the problem with today's academia very poignantly: "being successful in science requires developing Stockholm syndrome towards your superiors and cynicism towards your peers and juniors"

    Another commenter:
    "Science is a messy, messy place with morally bankrupt individuals competing with one another for paltry, waning grant funds. The only equity left in the world of science is social equity freely paid to influential professors who store it in a vault of ego.

    I dropped out of my (ivy league) PhD too. I refused to develop, as you say, Stockholm Syndrome for my advisors, who were using their students as robots to do their work instead of developing the students themselves. I once received a report that said something to the effect of “you desire to innovate, but that distracts you from your work.”

    There are almost NO checks and balances in the system. You can work for years under a professor who holds your degree or letter of recommendation hostage, and there is no way to earn your “payment” without delivering whatever expectation they have, no matter how absurd."

    In reaction to this letter a facebook group "Just science" was formed here:
    https://www.facebook.com/groups/586805534699776/

    There we find, that Nature recently has been active in this area.
    Here is a recent Nature special issue on the desolate conditions for budding scientists:
    http://www.nature.com/news/young-scientists-1.20835

    Want to read more stories? Nature asked young scientists to send in their stories and here they are: http://researchrealities.tumblr.com/

    Returning to the area of pure mathematics discussed in the article. The desolate situation might arguably be augmented in the field of pure maths, as all the best researchers in the US and UK work for the government. NSA for instance boasts, that it is the biggest employer of mathematicians in the world: https://twitter.com/nsacareers/status/550335064916389889
    Thus, those working in academia, with few exceptions, are simply those, who weren't good enough to get into government secret research.

    Now turning our gaze over to Japan, there, for historical reasons, the threeletter agencies are not braindraining academia as in the US. Thus, it might be expected, that the quality of the top academia (the best of the best of the best) is at a higher level in Japan than in the US. And that just aggravates the desolate situation in academia overall to the detriment of new groundbreaking ideas being understood, scrutinized and disseminated.

  • dear reader, we have moved on from this one peculiar situation to the complete rottenness of the academic system I see. I do not believe it is the time and place to deploy the frustrations of some PhD student (mathematician? theoretical physicist? who knows?) at a EUROPEAN institution (almost certainly nothing concrete to do with the peculiar problematic constellation discussed in this article), but I understand that this discussion has gotten as wide as the river Ganges and you want (as you certainly have tried from the start) to preach your preconceived notions about the things that are OH SO WRONG in this rotten system, never mind the people concerned too much.

    All this has made you forget most of common logic, when you pull bits of the article completely out of context: "90 percent of the people are working in small generality from a small number of cases in order to avoid the technical things that are done by the 10 percent minority" -> You realize this is said w.r.t. a situation where a lot of research in the field had moved on as far as it could without using the FO^3 tools which people could not use because there was a lack of understanding of what had been done by FO^3, in turn because there were still issues with the proofs that needed to be fixed, never mind that people tried very hard to understand/explain on both sides. Now McDuff and Wehrheim pointed out and helped with the fixing of the issues, thus helping the stifled flow of research around that area gain pace again. This is major community service in my books, when they certainly had worthwhile things of their own to work on, and yet you label them 'proofreaders' and what not, instead of acknowledging that they are helping to do good things to provide their community with tools to move research forward. That they did this will tell you they were good mathematicians, because you need a good mathematician to challenge a good mathematician's work and waterproof it.

    But go on missing the topic because you already know which mold everything in the world has to fit, go ahead.

  • Thank you Kevin Hartnett, for an engaging and well-explained article.
    But the real value was in the comments above, extremely illuminating, funny, and in places, cringe-inducing.
    Also many thanks to the rather balanced and elegant mathematicians who spent so much time clarifying areas of social complexity within their field – surprisingly effectively I thought, and very entertaining with it, particularly Chris Wendl.

  • An excellent article that explains how Fukaya's work was subsequently, and necessarily, elucidated with the help of McDuff and Wehrheim. This is how science and mathematics is supposed to work, and so it is ludicrous that discussions over relative merit should be politicised by references such as "women" and "asian".

  • Most has already been stated in this discussion. I will here try to summarize and add some information as to why we had to wait for 16 years for progress to be made on understanding the proof of Fukaya and Ono.

    The currency of science practiced today is impact ratings and citations. This is key to understand the siduation in academia today. Citations and impact are money and the game is thus to get these citations as a researcher gets grants and makes a career based on the number of citations he or he has in competition with others.

    Ph.D. students and tenure tracks are junior to their superiors who have the power to decide if they should be granted a degree, get recommendation letters and work. In order to survive in the academic environment, the supervisor can demand to be co-author, to have the Ph.D.s and tenure tracks do the work for him and to be citated when scrutinizing peer-revieved papers. He/she and can also form peer-review rings. This typically leads to breaking up one large paper into smaller, more shallow and less innovative papers in order to increase impact and citations.

    It is difficult to become wealthy in academics or just to get a mortgage and raise a family. When then the researcher has to chose between ethics (laws) and children, some chose their children and drop the ethics.

    Here are some papers discussing such academic malpractice:
    http://www.nature.com/news/watch-out-for-cheats-in-citation-game-1.20246
    http://scholarship.richmond.edu/cgi/viewcontent.cgi?article=1249&context=jolt
    http://ajpcell.physiology.org/content/295/3/C567.short#ref-11
    http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1017.3819&rep=rep1&type=pdf
    http://onlinelibrary.wiley.com/doi/10.1038/sj.embor.7401095/full
    https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1734216/
    https://scholarlykitchen.sspnet.org/2012/04/10/emergence-of-a-citation-cartel/
    http://www.pp.u-tokyo.ac.jp/wp-content/uploads/2016/03/GraSPP-DP-E-14-004-STIG-DP-E-14-004.pdf
    http://ajpcell.physiology.org/content/295/3/C567.short#ref-11
    http://journal.acs.org.au/index.php/ajis/article/download/1262/743

    Scientists with ground-breaking maths are not of any value, if they don't quote your work, rather they are competitors, who might persuade your of juniors to stop working for you.

    The situation of a Ph.D. and a tenure track in academia is more vulnerable than a similar position in the private sector, as there is no job security (see the commenter about the Stockholm syntrome in my previous post), no regulated work hours and the pay for little academics is very small. At the same time, the environment is very competitive.

    In the discussion to the person quitting his Ph.D. in optimization, (here: https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/) there are several commenters from the US agreeing with the contents of the letter, so it seems that this problem exists in the US too:
    https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/#comment-4781
    https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/#comment-4884
    https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/#comment-4913
    https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/#comment-4942

    And here is a mathematician commenting:
    https://crypto.junod.info/2013/09/09/an-aspiring-scientists-frustration-with-modern-day-academia-a-resignation/#comment-4983

    It is symptomatic, that the author of the biggest breakthrough in numerology last few years, Yitang Zhang and the twin prime conjecture didn't get any recommendation letters and thus couldn't work at the university with research. Talking about Nipping future Grothendiecks in the bud.

    It seems reasonable, that incentives for academic malpractice as discussed above will be relatively higher in an environment,where the best people in the country have gone working for the private sector and threeletter government agencies. It also seems logical, that our future Grothediecks will fail in such an academic environment, as it is not conductive to ground-breaking research, Yitang Zhang being a good example.

    The conditions of Ph.D. students and tenure tracks are discussed in the article “How Academia Resembles a Drug Gang”:
    “With a constant supply of new low-level drug sellers entering the market and ready to be exploited, drug lords can become increasingly rich without needing to distribute their wealth towards the bottom. You have an expanding mass of rank-and-file “outsiders” ready to forgo income for future wealth, and a small core of “insiders”  securing incomes largely at the expense of the mass. We can call it a winner-take-all market. […] The academic job market is structured in many respects like a drug gang, with an expanding mass of outsiders and a shrinking core  of insiders. Even if the probability that you might get shot in academia is relatively small (unless you mark student papers very harshly), one can observe similar dynamics. […] So what you have is an increasing number of brilliant PhD graduates arriving every year into the market hoping to secure a permanent position as a professor and enjoying freedom and high salaries, a bit like the rank-and-file drug dealer  hoping to become a drug lord. To achieve that, they are ready to forgo the income and security that they could have in other areas of employment by accepting insecure working conditions in the hope of securing jobs that are not expanding at the same rate. Because of the increasing inflow of potential outsiders ready to accept this kind of working conditions, this allows insiders to outsource a number of their tasks onto them, especially teaching, in a  context where there are increasing pressures for research and publishing.”

    Let us see how this harsh environment affects the students, who opt for the Ph.D.:
    “About 60% of graduate students said that they felt overwhelmed, exhausted, hopeless, sad, or depressed nearly all the time. One in 10 said they had contemplated suicide in the previous year.”
    http://www.sciencemag.org/careers/2014/02/paying-graduate-schools-mental-toll

    “A 2015 study at the University of California Berkeley found that 47% of graduate students suffer from depression […] A 2003 Australian study found that that the rate of mental illness in academic staff was three to four times higher than in the general population, according to a New Scientist article.
    https://qz.com/547641/theres-an-awful-cost-to-getting-a-phd-that-no-one-talks-about/

    “With half of all doctoral students leaving graduate school without finishing, something significant and overwhelming must be happening for at least some of them during the process of obtaining that degree. […]
    Scott Kerlin, a former doctoral-committee member at the University of Washington and the author of Pursuit of the Ph.D.: “Survival of the Fittest,” suggested that students describe the doctoral process as more “political” than intellectual in nature. There are “lots of issues of power and powerlessness that pervade the graduate experience,” Kerlin said, which may induce extreme distress for students who feels powerless. Indeed, a common reaction to highly stressful situations is difficulty engaging in mutual problem-solving, which, according to Rutledge, makes it especially important for graduate-school administrators to mediate discord between faculty.  
    But that can be hard to achieve: Many students are convinced the doctoral experience sets them up to fail. “Dysfunctional graduate departments, toxic faculty, and the Navy Seal-like brutality of the Ph.D. process all contribute to the burnout experienced by the estimated 50-plus percent of Ph.D. students who fail to earn their doctorates,” wrote Jill Yesko, then a doctoral student in geography, in a 2014 op-ed for Inside Higher Ed.“
    https://www.theatlantic.com/education/archive/2016/07/why-do-so-many-graduate-students-quit/490094/

    There are many ways to improve of this situation. The articles refrenced above and the comments under the letter of the Ph.D. quitter provides several constructive proposals. There are reasons to believe, that we can find a system to finance science, which aligns the goals of the researchers with scientific progress, forming an environment in which young Grothendiecks want to work and produce ground-breaking research.

  • This is a thorny question, but here is a constructive comment.
    One may distinguish between a hard won detailed and correct proof on the one hand and a less detailed but understandable proof based on previously hard won conceptual ground already gained by the mathematical community on the other.
    The style in analysis is very much to prefer the first case , while the possibilities of the second case is readily realized in topology, especially geometry related to algebraic topology.
    Starting from Poincare , to Alexander , to Lefschetz , to Whitney ,Thom, Milnor and beyond, the idea and meaning of a geometric object or some approximation thereof defining a homological object has emerged. It is something that mathematicians who have studied topology have assimilated.
    Those who are not comfortable this way would naturally want details and definitions.
    At the Simons Center two years ago roughly four groups came during the year to explain their methods to resolve the difficult matters in symplectic topology, which involve a passage from analysis of a set of partial differential equations
    and ostensibly their solution set to their homological implications in algebraic topology.
    There were two groups presenting the analysis side , very nice stuff.
    There were the presentations about kuranishi structures with one speaker using a good bit of the vague topological insight that I felt was perfectly sound but would not be understandable or comfortable to those not sufficiently indoctrinated .
    Finally, there was an abstract approach based on two categories with the at first glance outlandish claim " fibre products [ in this case transversal pull backs are such] cannot be defined without the extra structure of two-categories and anyone who studies this seriously will agree" Alas, I had already noticed in teaching algebraic topology that the basic concept of a fibration could not be adequately treated in the homotopy category per se but requires upping to a two-category
    so that the outlandish statement is most likely correct.

    So I am thinking everyone is right, we are all really trying to understand and I strongly recommend Bill Thurston's beautiful assay on this same debate from an earlier time.

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