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Michael Aschbacher and the sociology of mathematical proof w.r.t. the classification theorem for finite simple groups

Over dinner just now I had the pleasure of reading MIT PhD student Alma Steingart’s wonderful essay A group theory of group theory: collaborative mathematics and the ‘uninvention’ of a 1000-page proof, which is about the classification theorem for finite groups of simple order.

Steingart is one of those writers who highlights the very human enterprise that is sense-making in mathematics, including what counts as proof, which changes not only in time but even across different subfields. It’s an insight that first came to my attention during the hullabaloo over whether Shinichi Mochizuki actually did prove the abc conjecture back in 2012. It got really hammered home, however, in reading (and rereading) the late Bill Thurston’s classic On proof and progress in mathematics, especially the parts where he talks about both his introduction as a graduate student to the notion of proof-in-practice, and (much later) how the community responded to his proof of the hyperbolization theorem for Haken manifolds, in particular his realization that they were not interested in the thing proved itself so much as how he did it, his way of thinking.

This is echoed in Steingart’s abstract:

The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs.

Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise.

However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’. The papers that constituted it could still be found scattered throughout the mathematical literature, but no one other than the dwindling community of group theorists would know how to find them or how to piece them together.

Faced with this problem, finite group theorists resolved to produce a ‘second-generation proof’ to streamline and centralize the Classification. This project highlights that the proof and the community of finite simple groups theorists who produced it were co-constitutive–one formed and reformed by the other.

But I’ve harped on that at length elsewhere. Today I just want to collect some quotes on a monster mind.

Bill Thurston (hi again!) famously wrote that in his early years he inadvertently “killed the field” by being too good at proving results, driving others away, and that in retrospect this made him understand the sociology of mathematical proof better:

At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog.

An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.

I believe that two ecological effects were much more important in putting a damper on the subject than any exhaustion of intellectual resources that occurred.

First, the results I proved (as well as some important results of other people) were documented in a conventional, formidable mathematician’s style. They depended heavily on readers who shared certain background and certain insights. The theory of foliations was a young, opportunistic subfield, and the background was not standardized. I did not hesitate to draw on any of the mathematics I had learned from others. The papers I wrote did not (and could not) spend much time explaining the background culture. They documented top-level reasoning and conclusions that I often had achieved after much reflection and effort. I also threw out prize cryptic tidbits of insight, such as “the Godbillon-Vey invariant measures the helical wobble of a foliation”, that remained mysterious to most mathematicians who read them. This created a high entry barrier: I think many graduate students and mathematicians were discouraged that it was hard to learn and understand the proofs of key theorems.

Second is the issue of what is in it for other people in the subfield. When I started working on foliations, I had the conception that what people wanted was to know the answers. I thought that what they sought was a collection of powerful proven theorems that might be applied to answer further mathematical questions. But that’s only one part of the story. More than the knowledge, people want personal understanding. And in our credit-driven system, they also want and need theorem-credits.

I was thinking about what Thurston said above when I read about Michael Aschbacher in Steingart’s essay.

Michael Aschbacher (henceforth MA) was the second-most prolific contributor to the Classification effort, despite joining only towards the very end, with 21 papers to Daniel Gorenstein’s 25 (out of the 416 included in Gorenstein’s bibliography); DG, for context, was the closest the effort had to a leader (this also shows in the fact that he coauthored 15 papers with 6 others, over twice as many as the next-highest).

Here’s where the MA quotes start:

Though Michael Aschbacher became interested in finite simple groups only as a postdoc (having written his dissertation on combinatorics), his early contribution to the field took the community by surprise. Gorenstein described Aschbacher’s entrance into the field as ‘dramatic’ (Gorenstein, 1989: 469), and Solomon remarked that following Aschbacher’s work in 1974 and 1975, ‘belief mounted that nothing could stand between him and the completion of the Classification’ (Solomon, 2001: 338). In recognition of his many accomplishments, Aschbacher won the prestigious Frank Nelson Cole Prize in Algebra in 1980.

However, as Aschbacher’s esteem within the community grew, his
papers became notoriously difficult to comprehend. When Aschbacher’s first important paper in the field appeared in print, Solomon was an instructor in the Department of Mathematics at the University of Chicago. Together with George Glauberman, he tried to read the proof, but found it incredibly difficult to follow Aschbacher’s arguments. Specifically, Solomon complained that ‘there were these clever counting arguments, but he never bothered to write it [them] down’. Aschbacher’s style made it notoriously difficult for other mathematicians to follow his proofs, and as his papers grew longer, the difficulties were exacerbated.

Group theorists no longer judged a proof as a self-contained document, but took into account its author, his previous work, and his standing within the community. When asked about the process of refereeing in the finite simple group community, Gorenstein characterized Aschbacher’s papers as ‘extremely difficult’. He recalled, ‘Aschbacher is so smart … . Richard Lyons and I and Aschbacher had just written a joint paper … . I wrote a draft of this paper, Aschbacher rewrote it – I had enormous difficulty reading my own goddamn paper … . He can see things so much better than anybody else.’ He added, ‘he’s so quick that he misses things’.

If Gorenstein took on a leadership role as far as organization was concerned, Michael Aschbacher was the driving force behind completing the Classification. Beginning in 1973, Aschbacher proved one result after another, at least one of which had previously resisted sustained assaults by other mathematicians, including Thompson. When Aschbacher announced in Duluth that he had solved yet another problem, the group theorists in attendance were convinced that the end was in sight.

The Duluth conference had one more unexpected outcome. At the height of their excitement, group theorists began to realize that as their quest was coming to an end, so too was the field that had grown around it and the community that had sustained it. Mathematicians, especially the younger ones, started reevaluating whether they should stay in the field or move on to a new one. Duluth represented a turning point in the growth of the field. The sort of excitement that had characterized the field since the early 1960s was dampened as the end loomed on the horizon. Students were no longer drawn to the field, and while a complete Classification required several more years of sustained work, mathematicians had already laid claim to the major remaining problems.

From an interview with Ron Solomon:

Sometime around spring in ’75, Janko wrote to Danny and said, I give up. I can’t do the thin group problem … . Thompson who was visiting at Yale at the time started working on the thin group problem after Janko threw in the towel. He worked on it for several months around that spring and got stuck … . And I think Danny was seriously worried that if Janko couldn’t do it and if Thompson didn’t see at least after a few months of thinking about it how to finish it off, maybe there was some major intractable problem that the methods that we had developed just couldn’t handle. And Aschbacher picked up that problem over the summer and by the end of the summer finished it. And I think that was the psychological turning point of the
classification.

These quotes remind me of yet another monster mind — the late Jean Bourgain. This obituary by Terry Tao relates some of his experiences as a grad student struggling greatly with Bourgain’s papers, an experience common to most: https://terrytao.wordpress.com/2018/12/29/jean-bourgain/

Jeff Bezos

Weirdly enough, there’s been at least three different writers I enjoy reading who describe Jeff as “hyperintelligent”.

The first is Steve Yegge. From Amazon War Story #1: Jeff Bezos:

To prepare a presentation for Jeff, first make damn sure you know everything there is to know about the subject. Then write a prose narrative explaining the problem and solution(s). Write it exactly the way you would write it for a leading professor or industry expert on the subject.

That is: assume he already knows everything about it. Assume he knows more than you do about it. Even if you have groundbreakingly original ideas in your material, just pretend it’s old hat for him. Write your prose in the succinct, direct, no-explanations way that you would write for a world-leading expert on the material.

You’re almost done. The last step before you’re ready to present to him is this: Delete every third paragraph.

Now you’re ready to present!

Back in the mid-1800s there was this famous-ish composer/pianist named Franz Liszt. He is widely thought to have been the greatest sight-reader who ever lived. He could sight-read anything you gave him, including crazy stuff not even written for piano, like opera scores. He was so staggeringly good at sight-reading that his brain was only fully engaged on the first run-through. After that he’d get bored and start embellishing with his own additions.

Bezos is so goddamned smart that you have to turn it into a game for him or he’ll be bored and annoyed with you. That was my first realization about him. Who knows how smart he was before he became a billionaire — let’s just assume it was “really frigging smart”, since he did build Amazon from scratch. But for years he’s had armies of people taking care of everything for him. He doesn’t have to do anything at all except dress himself in the morning and read presentations all day long. So he’s really, REALLY good at reading presentations. He’s like the Franz Liszt of sight-reading presentations.

So you have to start tearing out whole paragraphs, or even pages, to make it interesting for him. He will fill in the gaps himself without missing a beat. And his brain will have less time to get annoyed with the slow pace of your brain.

I mean, imagine what it would be like to start off as an incredibly smart person, arguably a first-class genius, and then somehow wind up in a situation where you have a general’s view of the industry battlefield for ten years. Not only do you have more time than anyone else, and access to more information than anyone else, you also have this long-term eagle-eye perspective that only a handful of people in the world enjoy.

In some sense you wouldn’t even be human anymore. People like Jeff are better regarded as hyper-intelligent aliens with a tangential interest in human affairs.

But how do you prepare a presentation for a giant-brained alien? Well, here’s my second realization: He will outsmart you. Knowing everything about your subject is only a first-line defense for you. It’s like armor that he’ll eat through in the first few minutes. He is going to have at least one deep insight about the subject, right there on the spot, and it’s going to make you look like a complete buffoon.

Trust me folks, I saw this happen time and again, for years. Jeff Bezos has all these incredibly intelligent, experienced domain experts surrounding him at huge meetings, and on a daily basis he thinks of shit that they never saw coming. It’s a guaranteed facepalm fest.

So I knew he was going to think of something that I hadn’t. I didn’t know what it might be, because I’d spent weeks trying to think of everything. I had reviewed the material with dozens of people. But it didn’t matter. I knew he was going to blindside me, because that’s what happens when you present to Jeff.

(The quote ended up a bit longer than I thought, because I couldn’t resist Yegge’s entertaining style.)

The second is Brad Stone, who wrote The Everything Store. Quote:

Toward the end of the hour we spent discussing this book, Bezos leaned forward on his elbows and asked, “How do you plan to handle the narrative fallacy?” 

Ah yes, of course, the narrative fallacy. For a moment, I experienced the same sweaty surge of panic every Amazon employee over the past two decades has felt when confronted with an unanticipated question from the hyperintelligent
boss. The narrative fallacy, Bezos explained, was a term coined by Nassim Nicholas Taleb in his 2007 book The Black Swan to describe how humans are biologically inclined to turn complex realities into soothing but oversimplified stories. Taleb argued that the limitations of the human brain resulted in our species’ tendency to squeeze unrelated facts and events into cause-and-effect equations and then convert them into easily understandable narratives. These stories, Taleb wrote, shield humanity from the true randomness of the world, the chaos of human experience, and, to some extent, the unnerving element of luck  that plays into all successes and failures. Bezos was suggesting that Amazon’s rise might be that sort of impossibly  complex story. There was no easy explanation for how certain products were  invented, such as Amazon Web Services, its pioneering cloud business that so  many other Internet companies now use to run their operations. “When a  company comes up with an idea, it’s a messy process. There’s no aha moment,”  Bezos said. Reducing Amazon’s history to a simple narrative, he worried, could  give the impression of clarity rather than the real thing.  

To be fair, Brad also uses the same descriptive of Joy Covey, but I doubt anyone who knew her would disagree (Eugene Wei for instance, who was interviewed by both Jeff and Joy, puts them in the same sentence):

Another new arrival was Joy Covey as chief financial officer. Driven and often intimidating to underlings, Covey became an intellectual foil to Bezos and a key architect of Amazon’s early expansion. She had an unconventional background. A hyperintelligent but alienated child from San Mateo, California, she had run away from home when she was a sophomore in high school and worked as a grocery-store clerk in Fresno. She entered Cal State, Fresno, at age seventeen, graduated in two years, and then took the exam to become a certified public accountant at age nineteen, notching the second-highest score in the nation without studying. She later earned a joint business and law degree from Harvard. When Bezos found her, she was the thirty-three-year-old chief financial officer for a Silicon Valley digital-audio company called Digidesign. 

Over the next few years, Covey remained so intensely focused on executing  Bezos’s “get big fast” imperative that everything else in her life became  background noise. One morning she parked her car in the office garage and was  so distracted that she inadvertently left it running—all day. That evening, she  couldn’t find her car keys, concluded she had lost them, and went home without  her car. The security guard in the garage called her a few hours later and told her  that she might want to come back to the office to retrieve her still-idling vehicle. 

Another Brad quote on Jeff:

Right or wrong, Bezos’s behavior was often easier to accept because he was so frequently on target with his criticisms, to the amazement and often irritation of employees. Bruce Jones, the former Amazon vice president, describes leading a five-engineer team working to create algorithms to optimize pickers’  movements in the fulfillment centers while the company was trying to solve the  problem of batches. The group spent nine months on the task, then presented  their work to Bezos and the S Team. “We had beautiful documents and everyone  was really prepared,” Jones says. Bezos read the paper, said, “You’re all wrong,”  stood up, and started writing on the whiteboard.  

“He had no background in control theory, no background in operating  systems,” Jones says. “He only had minimum experience in the distribution  centers and never spent weeks and months out on the line.” But Bezos laid out  his argument on the whiteboard and “every stinking thing he put down was  correct and true,” Jones says. “It would be easier to stomach if we could prove  he was wrong but we couldn’t. That was a typical interaction with Jeff. He had  this unbelievable ability to be incredibly intelligent about things he had nothing  to do with, and he was totally ruthless about communicating it.” 

The third, most recent, is Eugene Wei, from Compress to impress:

Even with modern communication infrastructure, however, any modern CEO deals with amplification and distortion issues with any message. Humans learn about this problem very early on by playing telephone or operator, or what I just learned is more canonically known outside the U.S. as Chinese whispers. One person whispers a message in another person’s ear, and it’s passed on down the line to see if the original phrase can survive intact to the last person in the chain. Generally, errors accumulate along the way and what makes it to the end is some shockingly defective copy of the original.

Despite learning this lesson early on, most people in leadership positions still underestimate just how pervasive this problem is. This is why any manager or executive is familiar with how much time they spend on communicating the same things to different groups in the organization. It feels like it’s all you do sometimes, and yet you still encounter people who feel like they’re in the dark.

I hadn’t read Jeff Bezos’ most recent letter to shareholders until today, but it was just what I’d expect of it given something I observed in my seven years there, which are now more than a decade in the rear view mirror. In fact, one of reasons I hadn’t read it yet was that I suspected it would be very familiar, and it was. The other thing I suspected was that it would be really concise and memorable, and again, it was.

I suspect that very early on in his career as CEO, Jeff noticed the Chinese whispers problem as the company scaled. Anyone who is lucky enough to lead a successful company very quickly senses the impossibility of scaling one’s own time to all corners of the organization, but Jeff was laser focused on the more serious problem that presented, that of maintaining consistent strategy in all important decisions, many of which were made outside his purview each day. At scale, maintaining strategic alignment feels like an organizational design problem, but much of the impact of organizational design is centered around how it impacts information flow.

This problem is made more vexing by not just the telephone game issue but by the human inability to carry around a whole lot of directives in their minds. Jeff could spend a ton of time in All Hands meetings or with his direct reports and other groups inside Amazon, explaining his thinking in excruciating detail and hoping it sank in, but then he’d never have any time to do anything else.

Thankfully, humans have developed ways to ensure the integrity of messages persists across time when transmitted through the lossy mediums of oral tradition and hierarchical organizations.

One of these is to encode you message in a very distinctive format. There are many rhetorical tricks that have stood the test of time, like alliteration or anadiplosis. Perhaps supreme among these rhetorical forms is verse, especially when it rhymes. Both the rhythm and the rhyme (alliteration intentional) allow humans to compress and recall a message with greater accuracy than prose.

I never chatted with Bezos about this, so I don’t know if it was an explicit strategy on his part, but one of his great strengths as a communicator was the ability to encode the most important strategies for Amazon in very concise and memorable forms.

Go back even further, and there are dozens of examples of Bezos codifying key ideas for maximum recall. For example, every year I was at Amazon had a theme (reminiscent of how David Foster Wallace imagined in Infinite Jest that in the future corporate sponsors could buy the rights to name years). These themes were concise and memorable ways to help everyone remember the most important goal of the company that year.

One year, when our primary goal was to grow our revenue and order volume as quickly as possible to achieve the economies of scale that would capitalize on our high fixed cost infrastructure investments and put wind into our flywheel, the theme was “Get Big Fast Baby.” You can argue whether the “baby” at the end was necessary, but I think it’s more memorable with it than without. Much easier to remember than “Grow revenues 80%” or “achieve economies of scale” or something like that. …

I could continue on through the years, but what stands out is that I can recite these from memory even now, over a decade later, and so could probably everyone who worked at Amazon those years.

Here’s a good test of how strategically aligned a company is. Walk up to anyone in the company in the hallway and ask them if they know what their current top priority or mission is. Can they recite it from memory?

What Jeff understood was the power of rhetoric. Time spent coming up with the right words to package a key concept in a memorable way was time well spent. People fret about what others say about them when they’re not in the room, but Jeff was solving the issue of getting people to say what he’d say when he wasn’t in the room.

It was so important to him that we even had company-wide contests to come up with the most memorable ways to name our annual themes. One year Jeff announced at an All Hands meeting that someone I knew, Barnaby Dorfman, had won the contest. Jeff said the prize was that he’d buy something off the winner’s Amazon wish list, but after pulling Barnaby’s wish list up in front of the whole company on the screen, he said he didn’t think any of the items was good enough so instead he went over to the product page for image stabilized binoculars from Canon, retailing for over $1000, and bought those instead.

I have a list of dozens of Jeff sayings filed away in memory, and I’m not alone. It’s one reason he’s one of the world’s most effective CEO’s. What’s particularly impressive is that Jeff is so brilliant that it would be easy for him to state his thinking in complex ways that us mere mortals wouldn’t grok. But true genius is stating the complex simply.

Ironically, Jeff employs the reverse of this for his own information inflows. It’s well known that he banned Powerpoint at Amazon because he was increasingly frustrated at the lossy nature that medium. As Edward Tufte has long railed against, Powerpoint encourage people to reduce their thinking to a series of bullet points. Whenever someone would stand up in front of Jeff to present, Jeff would have rifled through to the end of the presentation before they would’ve finished a handful of slides, and Jeff would just jump in and start asking questions about slide 35 when someone was still talking to slide 3.

As a hyper intelligent person, Jeff didn’t want lossy compression or lazy thinking, he wanted the raw feed in a structured form, and so we all shifted to writing our arguments out as essays that he’d read silently in meetings. Written language is a lossy format, too, but it has the advantage of being less forgiving of broken logic flows than slide decks.

To summarize, Jeff’s outbound feedback was carefully encoded and compressed for maximum fidelity of transmission across hundreds of thousands of employees all over the world, but his inbound data feed was raw and minimally compressed. In structure, this pattern resembles what a great designer or photographer does. Find the most elegant and stable output from a complex universe of inputs.

(That quote ended up being long because I thought Eugene was very insightful throughout.)

Another Eugene Wei quote, from this article:

If Amazon has so many businesses that do make a profit, then why is it still showing quarterly losses, and why has even free cash flow decreased in recent years?

Because Amazon has boundless ambition. It wants to eat global retail. This is one area where the press and pundits accept Amazon’s statements at face value.

Given that giant mission, Amazon has decided to continue to invest to arm itself for a much larger scale of business. If it were purely a software business, its fixed cost investments for this journey would be lower, but the amount of capital required to grow a business that has to ship millions of packages to customers all over the world quickly is something only a handful of companies in the world could even afford. Joey Chestnut doesn’t just wake up one day and win the Coney Island hot dog eating contest every year, he has to spend months of training to prepare his digestive system for the feat.

Amazon has seen that lowering its shipping costs and increasing the speed of shipping items to customers is like a shot of adrenaline to customers’ propensity to buy from them, and so it has doubled down on building more and more fulfillment centers around the world. When I joined Amazon it had one fulfillment center. Today it has dozens just in the U.S. alone, and I would not be surprised if it has more than 100 fulfillment centers worldwide now.

That is a gargantuan investment, billions of dollars’ worth, and it takes a significant bite out of Amazon’s free cash flow. Add in its investments in infrastructure to support a growing AWS client base, and Amazon has again hiked its fixed cost base to a higher plateau. But for Amazon this is nothing new, it’s just the same typeface bolded.

I’m convinced Amazon could easily turn a quarterly profit now. Many times in its history, it could have been content to stop investing in new product lines, new fulfillment centers, new countries. The fixed cost base would flatten out, its sales would continue growing for some period of time and then flatten out, and it would harvest some annuity of profits. Even the first year I joined Amazon in 1997, when it was just a domestic book business, it could have been content to rest on its laurels.

But Jeff is not wired that way. There are very few people in technology and business who are what I’d call apex predators. Jeff is one of them, the most patient and intelligent one I’ve met in my life. An apex predator doesn’t wake up one day and decide it is done hunting. Right now I envision only one throttle to Jeff’s ambitions and it is human mortality, but I would not be surprised if one day he announced he’d started another side project with Peter Thiel to work on a method of achieving immortality.

And now for a complete tangent.

Reading about Bezos and Musk in Tim Fernholz’s Rocket Billionaires book makes me feel like my childhood dream of going to space may just become true before I die, due to the these guys’ efforts to make it semi-affordable. I don’t mind saving for years for the chance to go on such a trip.

I also like that Bezos is a fan of O’Neill cylinders:

Bezos is unabashed in his fanaticism for Star Trek and its many spin-offs. He has a holding company called Zefram, which honors the character who invented warp drive. He persuaded the makers of the film Star Trek Beyond to give him a cameo as a Starfleet official. He named his dog Kamala, after a woman who appears in an episode as Picard’s “perfect” but unattainable mate. As time has passed, Bezos and Picard have physically converged. Like the interstellar explorer, portrayed by Patrick Stewart, Bezos shaved the remnant strands on his high-gloss pate and acquired a cast-iron physique. A friend once said that Bezos adopted his strenuous fitness regimen in anticipation of the day that he, too, would journey to the heavens.

When reporters tracked down Bezos’s high-school girlfriend, she said, “The reason he’s earning so much money is to get to outer space.” This assessment hardly required a leap of imagination. As the valedictorian of Miami Palmetto Senior High School’s class of 1982, Bezos used his graduation speech to unfurl his vision for humanity. He dreamed aloud of the day when millions of his fellow earthlings would relocate to colonies in space. A local newspaper reported that his intention was “to get all people off the Earth and see it turned into a huge national park.”

Most mortals eventually jettison teenage dreams, but Bezos remains passionately committed to his, even as he has come to control more and more of the here and now. Critics have chided him for philanthropic stinginess, at least relative to his wealth, but the thing Bezos considers his primary humanitarian contribution isn’t properly charitable. It’s a profit-seeking company called Blue Origin, dedicated to fulfilling the prophecy of his high-school graduation speech. He funds that venture—which builds rockets, rovers, and the infrastructure that permits voyage beyond the Earth’s atmosphere—by selling about $1 billion of Amazon stock each year. More than his ownership of his behemoth company or of The Washington Post—and more than the $2 billion he’s pledged to nonprofits working on homelessness and education for low-income Americans—Bezos calls Blue Origin his “most important work.”

He considers the work so important because the threat it aims to counter is so grave. What worries Bezos is that in the coming generations the planet’s growing energy demands will outstrip its limited supply. The danger, he says, “is not necessarily extinction,” but stasis: “We will have to stop growing, which I think is a very bad future.” While others might fret that climate change will soon make the planet uninhabitable, the billionaire wrings his hands over the prospects of diminished growth. But the scenario he describes is indeed grim. Without enough energy to go around, rationing and starvation will ensue. Over the years, Bezos has made himself inaccessible to journalists asking questions about Amazon. But he shares his faith in space colonization with a preacher’s zeal: “We have to go to space to save Earth.”

At the heart of this faith is a text Bezos read as a teen. In 1976, a Princeton physicist named Gerard K. O’Neill wrote a populist case for moving into space called The High Frontier, a book beloved by sci-fi geeks, NASA functionaries, and aging hippies. As a Princeton student, Bezos attended O’Neill seminars and ran the campus chapter of Students for the Exploration and Development of Space. Through Blue Origin, Bezos is developing detailed plans for realizing O’Neill’s vision.

The professor imagined colonies housed in miles-long cylindrical tubes floating between Earth and the moon. The tubes would sustain a simulacrum of life back on the mother planet, with soil, oxygenated air, free-flying birds, and “beaches lapped by waves.” When Bezos describes these colonies—and presents artists’ renderings of them—he sounds almost rapturous. “This is Maui on its best day, all year long. No rain, no storms, no earthquakes.” Since the colonies would allow the human population to grow without any earthly constraints, the species would flourish like never before: “We can have a trillion humans in the solar system, which means we’d have a thousand Mozarts and a thousand Einsteins. This would be an incredible civilization.” Bezos rallies the public with passionate peroration and convincing command of detail. 

Hollywood accounting and cost disease in medicine (also Stan Lee)

I always wondered what people meant when they (usually obliquely/snidely) referred to “creative accounting practices”. Today I stumbled upon an example entirely by happenstance, called “Hollywood accounting”, in a comment of jimrandomh‘s on his LW container for short-form writing:

The discussion so far on cost disease seems pretty inadequate, and I think a key piece that’s missing is the concept of Hollywood Accounting. Hollywood Accounting is what happens when you have something that’s extremely profitable, but which has an incentive to not be profitable on paper. The traditional example, which inspired the name, is when a movie studio signs a contract with an actor to share a percentage of profits; in that case, the studio will create subsidiaries, pay all the profits to the subsidiaries, and then declare that the studio itself (which signed the profit-sharing agreement) has no profits to give.

In the public contracting sector, you have firms signing cost-plus contracts, which are similar; the contract requires that profits don’t exceed a threshold, so they get converted into payments to de-facto-but-not-de-jure subsidiaries, favors, and other concealed forms. Sometimes this involves large dead-weight losses, but the losses are not the point, and are not the cause of the high price.

In medicine, there are occasionally articles which try to figure out where all the money is going in the US medical system; they tend to look at one piece, conclude that that piece isn’t very profitable so it can’t be responsible, and move on. I suspect this is what’s going on with the cost of clinical trials, for example; they aren’t any more expensive than they used to be, they just get allocated a share of the profits from R&D ventures that’re highly profitable overall.

Jim fleshes out the last remark:

I’m pretty uncertain how the arrangements actually work in practice, but one possible arrangement is: You have two organizations, one of which is a traditional pharmaceutical company with the patent for an untested drug, and one of which is a contract research organization. The pharma company pays the contract research organization to conduct a clinical trial, and reports the amount it paid as the cost of the trial. They have common knowledge of the chance of success, of the future probability distribution of future revenue for the drug, how much it costs to conduct the trial, and how much it costs to insure away the risks. So the amount the first company pays to the second is the costs of the trial, plus a share of the expected profit.

Pharma companies making above-market returns are subject to political attack from angry patients, but contract research organizations aren’t. So if you control both of these organizations, you would choose to allocate all of the profits to the second organization, so you can defend yourself from claims of gouging by pleading poverty.

Stan Lee is a particularly high-profile victim of Hollywood accounting. Per the Wiki article:

Stan Lee, co-creator of the character Spider-Man, had a contract awarding him 10% of the net profits of anything based on his characters. The film Spider-Man (2002) made more than $800 million in revenue, but the producers claim that it did not make any profit as defined in Lee’s contract, and Lee received nothing. In 2002 he filed a lawsuit against Marvel Comics. The case was settled in January 2005, with Marvel paying $10 million to “finance past and future payments claimed by Mr. Lee.”

(Wow, that makes me angry. How on earth could you do that to Stan Lee?)

Peter Jackson is another:

Peter Jackson, director of The Lord of the Rings, and his studio Wingnut Films, brought a lawsuit against New Line Cinema after an audit. Jackson stated this is regarding “certain accounting practices”. In response, New Line stated that their rights to a film of The Hobbit were time-limited, and since Jackson would not work with them again until the suit was settled, he would not be asked to direct The Hobbit, as had been anticipated.[18] Fifteen actors are suing New Line Cinema, claiming that they have never received their 5% of revenue from merchandise sold in relation to the movie, which contains their likenesses.[19] Similarly, the Tolkien estate sued New Line, claiming that their contract entitled them to 7.5% of the gross receipts of the $6 billion hit.[20] According to New Line’s accounts, the trilogy made “horrendous losses” and no profit at all.

The Harry Potter franchise is also another:

A WB receipt was leaked online, showing that the hugely successful movie Harry Potter and the Order of the Phoenix ended up with a $167 million loss on paper.[23] This is especially egregious given that, without inflation adjustment, the Harry Potter film series is the second highest-grossing film series of all time both domestically and internationally, second only to the Marvel Cinematic UniverseHarry Potter and the Deathly Hallows – Part 2 remains highest-grossing movie ever for Warner Bros.

This is all baffling to me, even sans ethics — why would you overdo Hollywood accounting on such visible projects? A slightly-less-stupid villain would do less-than-extreme creative accounting on lots of lower-profile revenue-optimized films (“diversifying their portfolio”), and hence be able to fly under the radar longer. Of course there probably are such syndicates, and they are still flying under the radar; we just see the stupid ones – this is embarrassingly obvious in retrospect.

Amusingly, creative accounting is a central plot device in Mel Brooks’ film The Producers, whose Wiki summary sounds amazing.

The Producers (1968).jpg

P vs NP humor

Parody of a typical comp.theory newsgroup discussion of a typical P vs NP proof, from Suresh Venkatasubramanian’s post A meta-proof:

P: I would like to announce my proof of P=/!=NP. The proof is very short and demonstrates how to solve/not solve SAT in polynomial time. You may find a write up of the proof here.

|– V: I started reading your proof and when you claim ‘foobar’ do you mean ‘foobar’ or ‘phishbang’ ?
|—-P: I meant ‘phishbang’. Thanks for pointing that out. An updated version is here.
|——V: Well if you meant ‘phishbang’ then statement “in this step we assume the feefum” is incorrect.
|——–P: No no, you don’t understand. I can assume feefum because my algorithm has a glemish.
|———–V: It has a glemish ? !! But having a glemish doesn’t imply anything. All algorithms have glemishes !!
|—-V’: Yes, and in fact in the 3rd step of argument 4, your glemish contradicts the first propum.
|–V”: I think you need to understand some basic facts about complicity theory before you can go further. Here is a book to read.
|—-P: My proof is quite clear, and I don’t see why I have to explain it to you if you don’t understand. I have spent a long time on this.
|——V’: Um, this is a famous problem, and there are many false proofs, and so you do have to convince us that the argument using glemishes can actually work.
|——–P: But what is wrong in my proof ? I don’t see any problems with it, and if you can’t point one out, how can you say it is wrong.
|———-V””: I don’t have to read the entire proof: glemished algorithms are well known not to work.
|————V”””: Check out this reference by to see why.
P: <silence>
|–P: <answering earlier post>. This is what I mean by a glemish. it is really a flemish, not a glemish, which answers your objection.
|—-P’: Keep up the good work P. I tried publishing my result, and these people savaged my proof without even trying to identify a problem. All great mathematical progress has come from amateurs like us. See this link of all the theorems proved by non-experts.
|——V’: Oh jeez, not P’ again. I thought we had established that your proof was wrong.
|——–P’: no you didn’t: in fact I have a new version that explains the proof in such simple language even dumb&%&%s like you can get it.
|——P: Thanks P’, I understand that there will be resistance from the community since I have proved what they thought to be so hard.
|–V’: P, I’m trying to understand your proof, with the flemishes, and it seems that maybe there is a problem in step 517 with the brouhaha technique.
P: <silence>
|—-P: V’, thanks for pointing out that mistake. you are right. Instead of a brouhaha technique I need a slushpit. The details are complicated, so I will fix it and post a corrected version of the proof shortly. Thanks to all those who gave me constructive advice. I am glad that at least some of you have an open mind to accept new ideas.

The Peter Scholze experience

There are lots of rising stars in math. There are a number of superstars too. (Exercise for the interested reader: browse through the laudatios for all the recent high-profile prizes, see what names pop up again and again.)

But every once in a while comes a person to whom others feel called. Think Grothendieck. The word I have in mind is “prophet”, although that seems a bit strong to use for just about anyone else other than Schurik (what his close friends called him). This person isn’t a star so much as a phenomenon, an experience.

(Even Terry isn’t “an experience” in the sense I’m gesturing towards, although he’s my favorite contemporary mathematician, given my penchant for generalists and master expositors. It’s rare that you find someone who’s both at the highest levels. I’d call him, not a prophet, but a bridge-builder, a weaver of worlds. But I digress.)

Everything I’ve read about Peter Scholze makes me think he’s a prophet in the style of Schurik. Here are some quotes to that effect.

From redditor Professor_Pohato, whose assessment is wonderfully prosaic:

Maths at [the Free University of Bonn] is no joke, this faculty is considered on of the finest and hardest in town which makes it even more impressive he pulled it off in under two years.

Kudos to Peter and his beautiful hair

From Ken Ribet’s interview on the occasion of him winning the 2017 Brouwer Medal:

Let’s talk about recent developments in the field. Is there something that gets you really excited? If you would have two years to study only something mathematical, what would it be?

Well, it would clearly be perfectoid spaces.

I actually wrote that down as next subject! Two of our Utrecht students attended the Arizona Winter School about that, I guess you were not there?

No, but their inventor Peter Scholze was in Berkeley for a semester fairly recently and he gave a course. I have tried to follow the beginning of the course and I was just completely blown away by (1) the beauty of the subject and (2) the amazing command of an entire landscape that he has. He was lecturing to many people who were experts in different corners, like (, z C) -modules and different ways of expressing Galois representations in somewhat concrete terms and Scholze understands all of this in his new framework in some illuminating way that completely surprised the original perpetrators of the subject, who were sitting in the audience shaking their heads. …

From Spiegel Online’s interview with Klaus Altmann, Scholze’s former teacher at the Free University of Berlin, who he generously describes as “an important mentor”:

SPIEGEL ONLINE: Peter Scholze has described you as an important mentor. How did you meet him?

Altmann: Scholze visited as a talent a math circle at his special school in Berlin. The head of the class told me that she has a really great student, to whom she simply can not cope, and who needs further suggestions. Then she sent him to me at the FU. Scholze was 16 years old at the time.

SPIEGEL ONLINE: How did you experience him?

Altmann: We talked a bit – and then I gave him a textbook for students and told him to look at the first one or two chapters. We could then ask questions at the next meeting. After two weeks, he came to me and read the whole book. He explained to me what was not quite okay in the book and what to do better. At the age of 16 — that was really amazing.

SPIEGEL ONLINE: Did Scholze study with you then?

Altmann: No, that’s not the way to say that – we just kept in touch. After our first conversation, I took him to my research seminar. Then I asked him how much he understood. He meant not much. Because he did not know many basic terms that were used in the seminar lecture. I explained most of those terms in a few minutes.”Now everything is clear,” he said suddenly. He must have completely saved those 90 minutes before he could understand them. And after the gaps were filled, he could retrieve and assemble everything. For the first time, I got an idea of ​​how he processes things. Scholze may have something like a photographic memory for math, he hardly takes notes.

SPIEGEL ONLINE: How did the mini-study by Scholze run for you?

Altmann: He was there almost every week. He came to my algebra seminar and heard lectures there and gave lectures to my graduate students and doctoral students. The fact that a 16-year-old taught and explained something to them was psychologically not always easy for the 25-year-old students.

SPIEGEL ONLINE: And how did Scholze deal with this situation?

Altmann: It did not seem to be a special situation for him. Not only is he an exceptional mathematician – he is also a very impressive personality. He explained things so that the 25-year-olds were not piqued, but enthusiastic. Scholze had an exact idea of ​​what the others knew and what did not and how he had to explain something so they could understand it. That was completely natural for him and incredibly fascinating to me. I have never experienced such a person.

SPIEGEL ONLINE: Would not Scholze get bored with you at some point?

Altmann: I hope not. But he is also far from any arrogance and would not let me feel that. In his view, the students and even the lecturers and professors should actually all be non-professionals because he is intellectually so far ahead of them. But he does not think so – he is completely down to earth. The cliché of a mundane, introverted mathematician may be true to some – it is totally wrong with him.

SPIEGEL ONLINE: What makes Scholze so special as a mathematician?

Altmann: He oversees complicated problems with incredible ease. And he has a crazy mind. There is a standard book on algebraic geometry, which I have given Scholze. It’s our Bible, so to speak. Even for many doctoral students it is difficult to work through this text completely. Scholze told me that he read the book during German lessons. And reading means understanding with him then.

SPIEGEL ONLINE: But you as a math professor can do it.

Altmann: When I’m working on an unknown math book, I read a line or two, put the book aside, take a pen, do a bit of calculation, and see what that means. And after half a day, in a nutshell, I read on the third line. The density of information in math texts is very high. Scholze, on the other hand, just reads it like an entertaining novel. But not superficially, he understands it to the last detail. One could say: Mathematics is his second native language, his brain can directly record the complicated statements without tedious processing and translation. But that does not mean that he only has mathematics in mind. Scholze has also graduated from high school.

When Altmann mentions “our Bible” (of algebraic geometry) I think he’s talking about Hartshorne. Interestingly, Hartshorne is doubly-distilled/simplified EGA, at least if you believe Elencwajg on MO[1].

Also funny is Altmann’s argument that Peter “doesn’t just have math in mind – he’s also graduated from high school”. It seems an exceedingly low bar to clear, but then again there are the Ramanujans of the world, so what do I know w.r.t. generational minds like theirs?

From Allyn Jackson’s Fields laudatio for Scholze:

Peter Scholze possesses a type of mathematical talent that emerges only rarely. He has the capacity to absorb and digest the entire frontier of a broad swath of mathematical research ranging over many diverse and often inchoate developments. What is more, he sees how to integrate these developments through stunning new syntheses that bring forth the simplicity that had previously been shrouded. Much of his work is highly abstract and foundational, but it also exhibits a keen sense of exactly which new concepts and techniques will enable proofs of important concrete results.

It was at a 2011 conference that Scholze, then still a doctoral student, first described the concept of perfectoid spaces, thereby setting off a revolution in algebraic and arithmetic geometry. The concept was quickly embraced by researchers the world over as just the right notion to clarify a wide variety of phenomena and shed new light on problems that had evaded solution for decades. …

Scholze is not simply a specialist in p-adic mathematics for a fixed p. For instance, he has recently been developing a sweeping vision of a “universal” cohomology that works over any field and over any space. In the 1960s, Grothendieck described his theory of motives, the goal of which was to build such a universal cohomology theory. While Vladimir Voevodsky (2002 Fields Medalist) made significant advances in developing the theory of motives, for the most part Grothendieck’s vision has gone unfulfilled. Scholze is coming at the problem from the other side, so to speak, by developing an explicit cohomology theory that in all observable ways behaves like a universal cohomology theory. Whether this theory fulfills the motivic vision then becomes a secondary question. Mathematicians the world over are following these developments with great excitement.

The work of Peter Scholze is in one sense radically new, but in another sense represents an enormous expansion, unification, and simplification of ideas that were already in the air. It was as if a room were in semi-darkness, with only certain corners illuminated, when Scholze’s work caused the flip of a light switch, revealing in bright detail the features of the room. The effect was exhilarating if rather disorienting. Once mathematicians had adjusted to the new light, they began applying the perfectoid viewpoint to a host of outstanding problems.

The clarity of Scholze’s lectures and written expositions played a large role in making the prospect of joining the perfectoid adventure appear so attractive to so many mathematicians, as has his personality, universally described as kind and generous.

From ICM 2018’s blog post:

Peter Scholze was a massive hit at this year’s ICM, and was a favorite to win the Fields Medal months before the winners were announced on Wednesday 1st August. His plenary lecture on ‘Period Maps in p-adic Geometry’ was so popular, Riocentro staff had to open additional seating in Pavilion 6 on Saturday morning. Everyone wanted to see the young mathematician in action.

The Jimi Hendrix of mathematics took to the stage to explain his breathtaking and groundbreaking work in the field of geometry using his own, homemade, handwritten slides. …

Scholze’s work has stunned the math community since his early twenties at Bonn University in Germany. “Peter’s work has really completely transformed what can be done, what we have access to,” said collaborator Ana Caraiani.

From Michael Rapoport’s laudatio of Scholze’s Fields medal:

Scholze has proved a whole array of theorems in p-adic geometry. These theorems are not disjoint but, rather, are the outflow of a theoretical edifice that Scholze has created in the last few years. There is no doubt that Scholze’s ideas will keep mathematicians busy for many years to come.

What is remarkable about Scholze’s approach to mathematics is the ultimate simplicity of his ideas. Even though the execution of these ideas demands great technical power (of which Scholze has an extraordinary command), it is still true that the initial key idea and the final result have the appeal of inevitability of the classics, and their elegance. We surely can expect more great things of Scholze in the future, and it will be fascinating to see to what further heights Scholze’s work will take him.

From Quanta in 2016:

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem … Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually—I only assimilated it through learning some other stuff.” …

After high school, Scholze continued to pursue this interest in number theory and geometry at the University of Bonn. In his mathematics classes there, he never took notes, recalled Hellmann, who was his classmate. Scholze could understand the course material in real time, Hellmann said. “Not just understand, but really understand on some kind of deep level, so that he also would not forget.” …

Scholze “found precisely the correct and cleanest way to incorporate all the previously done work and find an elegant formulation for that—and then, because he found really the correct framework, go way beyond the known results,” Hellmann said. …

Despite the complexity of perfectoid spaces, Scholze is known for the clarity of his talks and papers. “I don’t really understand anything until Peter explains it to me,” Weinstein said.

Scholze makes a point of trying to explain his ideas at a level that even beginning graduate students can follow, Caraiani said. “There’s this sense of openness and generosity in terms of ideas,” she said. “And he doesn’t just do that with a few senior people, but really, a lot of young people have access to him.” Scholze’s friendly, approachable demeanor makes him an ideal leader in his field, Caraiani said. One time, when she and Scholze were on a difficult hike with a group of mathematicians, “he was the one running around making sure that everyone made it and checking up on everyone,” Caraiani said.

Yet even with the benefit of Scholze’s explanations, perfectoid spaces are hard for other researchers to grasp, Hellmann said. “If you move a little bit away from the path, or the way that he prescribes, then you’re in the middle of the jungle and it’s actually very hard.” But Scholze himself, Hellmann said, “would never lose himself in the jungle, because he’s never trying to fight the jungle. He’s always looking for the overview, for some kind of clear concept.”

Scholze avoids getting tangled in the jungle vines by forcing himself to fly above them: As when he was in college, he prefers to work without writing anything down. That means that he must formulate his ideas in the cleanest way possible, he said. “You have only some kind of limited capacity in your head, so you can’t do too complicated things.” …

Discussing mathematics with Scholze is like consulting a “truth oracle,” according to Weinstein. “If he says, ‘Yes, it is going to work,’ you can be confident of it; if he says no, you should give right up; and if he says he doesn’t know—which does happen—then, well, lucky you, because you’ve got an interesting problem on your hands.”

Yet collaborating with Scholze is not as intense an experience as might be expected, Caraiani said. When she worked with Scholze, there was never a sense of hurry, she said. “It felt like somehow we were always doing things the right way—somehow proving the most general theorem that we could, in the nicest way, doing the right constructions that will illuminate things.” …

Scholze continues to explore perfectoid spaces, but he has also branched out into other areas of mathematics touching on algebraic topology, which uses algebra to study shapes. “Over the course of the last year and a half, Peter has become a complete master of the subject,” Bhatt said. “He changed the way [the experts] think about it.”

It can be scary but also exciting for other mathematicians when Scholze enters their field, Bhatt said. “It means the subject is really going to move fast. I’m ecstatic that he’s working in an area that’s close to mine, so I actually see the frontiers of knowledge moving forward.”

Yet to Scholze, his work thus far is just a warm-up. “I’m still in the phase where I’m trying to learn what’s there, and maybe rephrasing it in my own words,” he said. “I don’t feel like I’ve actually started doing research.”

“I’m just phrasing things in my own words” is exceedingly modest. Rephrasings by the masters can be so illuminating as to alter the research directions of entire subfields. Consider John Milnor, legendarily precocious youngster turned all-time great, probably the most decorated mathematician alive. When he decided to cast his eye on dynamical systems theory in the 1970s, “the Smale program in dynamics had been completed”, per Peter Makienko in his review of Topological Methods in Modern Mathematics. No worries, Milnor said, I’m just trying to teach myself. Peter wrote:

Milnor’s approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré’s work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor’s work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.

(He did have the advantage of collaborating with the greatest intuitive geometric thinker in the history of mathematics in Thurston, but the point stands.)

But I digress again. Back to Peter. Last quotes, from Michael Harris’ The perfectoid concept: test case for an absent theory:

It’s not often that contemporary mathematics provides such a clear-cut example of concept formation as the one I am about to present: Peter Scholze’s introduction of the new notion of perfectoid space.

The 23-year old Scholze first unveiled the concept in the spring of 2011 in a conference talk at the Institute for Advanced Study in Princeton. I know because I was there. This was soon followed by an extended visit to the Institut des Hautes Études Scientifiques (IHES) at Bûressur-Yvette, outside Paris — I was there too. Scholze’s six-lecture series culminated with a spectacular application of the new method, already announced in Princeton, to an outstanding problem left over from the days when the IHES was the destination of pilgrims come to hear Alexander Grothendieck, and later Pierre Deligne, report on the creation of the new geometries of their day.

Scholze’s exceptionally clear lecture notes were read in mathematics departments around the world within days of his lecture — not passed hand-to-hand as in Grothendieck’s day — and the videos of his talks were immediately made available on the IHES website. Meanwhile, more killer apps followed in rapid succession in a series of papers written by Scholze, sometimes in collaboration with other mathematicians under 30 (or just slightly older), often alone.
By the time he reached the age of 24, high-level conference invitations to talk about the uses of perfectoid spaces (I was at a number of those too) had enshrined Scholze as one of the youngest elder statesmen ever of arithmetic geometry, the branch of mathematics where number theory meets algebraic geometry.) Two years later, a week-long meeting in 2014 on Perfectoid Spaces and Their Applications at the Mathematical Sciences Research Institute in Berkeley broke all attendance records for “Hot Topics” conferences. …

Four years after its birth, perfectoid geometry, the theory of perfectoid spaces, is a textbook example of a progressive research program in the Lakatos sense. It is seen, retrospectively, as the right theory toward which several strands of arithmetic geometry were independently striving. It has launched a thousand graduate student seminars (if I were a historian I would tell you exactly how many); the students’ advisors struggle to keep up. It has a characteristic terminology, notation, and style of argument; a growing cohort of (overwhelmingly) young experts, with Scholze and his direct collaborators at the center; a domain of applications whose scope continues to expand to encompass new branches of mathematics; an implicit mandate to unify and simplify the fields in its immediate vicinity.

Last, but certainly not least, there is the generous, smiling figure of Peter Scholze himself, in the numerous online recordings of his lectures or in person, patiently answering every question until his questioner is satisfied, still just 27 years old, an inexhaustible source of revolutionary new ideas. …

I’m not the only person who thinks that the Scholze experience is reminiscent of the Schurik experience 🙂 I also like the imagery of Scholze at the center surrounded by his ever-growing rings of collaborators, subsuming and simplifying and unifying everything they touch (even though the usual parallels in sci-fi are meant to be horrifying).

What does Harris mean by “clear-cut example of concept formation”?

“Category” is the formalized mathematical concept that currently best captures what is understood by the word “concept.” Scholze defined perfectoid spaces as a category of geometric spaces with all the expected trappings, and thus there’s no reason to deny it the status of “concept.” I will fight the temptation to explain in any more detail just why Scholze’s perfectoid concept was seen to be the right one as soon as he explained the proofs in the (symbolically charged) suburban setting of the IHES. But I do want to disabuse the reader of any hope that the revelation was as straightforward as a collective process of feeling the scales fall from our eyes. Scholze’s lectures and expository writing are of a rare clarity, but they can’t conceal the fact that his proofs are extremely subtle and difficult. Perfectoid rings lack familiar finiteness properties — the term of art is that they are not noetherian. This means that the unwary will be systematically led astray by the familiar intuitions of algebra. The most virtuosic pages in Scholze’s papers generally involve finding ways to reduce constructions that appear to be hopelessly infinite to comprehensible (finite type) ring theory.

A few months after his IHES lectures, a French graduate student asked whether I would be willing to be his thesis advisor; things started conventionally enough, but very soon the student in question was bitten by the perfectoid bug and produced a Mémoire M2 — a mainly expository paper equivalent to a minor thesis — that was much too complicated for his helpless advisor. By then Scholze had found two new spectacular applications that the precocious student managed to cram into his Mémoire M2, making it by far the longest Mémoire it has even been my pleasure to direct. (The student in question — who has taken on by a second, more competent advisor — has not yet finished his thesis but I would already count him as a member of the second perfectoid circle revolving around Scholze. The first circle, as I see it, includes Scholze’s immediate collaborators and a few others; the second circle is already much broader, and there is a third circle consisting of everyone hoping to apply the concept to one thing or another.) …

More of that rings imagery!

I began writing this essay three years after Scholze’s IHES lectures and one month after his ICM lecture in Seoul. One year earlier, I could safely assert that no one had (correctly) made use of the perfectoid concept except in close collaboration with Scholze. The Seoul lecture made it clear that this was already no longer the case. Now, after nearly a year has passed, the perfectoid concept has been assimilated by the international community of arithmetic geometers and a growing group of number theorists, in applications to questions that its creator had never considered. It is an unqualified success.

How can this be explained? Mathematicians in fields different from mine are no better prepared than philosophers or historians to evaluate our standards of significance. One does occasionally hear dark warnings about disciplines dominated by cliques that expand their influence by favorably reviewing one another’s papers, but by and large, when a field as established and prestigious as arithmetic geometry asserts unanimously that a young specialist is the best one to come along in decades, our colleagues in other fields defer to our judgment.

Doubts may linger nonetheless. I don’t think that even a professional historian would see the point in questioning whether Scholze is exceedingly bright, but is his work really that important? How much of the fanfare around Scholze is objectively legitimate, how much an effect of Scholze’s obvious brilliance and unusually appealing personality, and how much just an expression of the wish to have something to celebrate, the “next big thing”? Is a professional historian even allowed to believe that (some) value judgments are objective, that the notion of the right concept is in any way coherent? How can we make sweeping claims on behalf of perfectoid geometry when historical methodology compels us to admit that even complex numbers may someday be seen as a dead end? “Too soon to tell,” as Zhou En-Lai supposedly said when asked his opinion of the French revolution.

It’s possible to talk sensibly about convergence without succumbing to the illusion of inevitability. In addition to the historical background sketched above, and the active search for the right frameworks that many feel Scholze has provided, perfectoid geometry develops themes that were already in the air when Scholze began his career. With respect to the active research programs that provide a field with its contours, it’s understandable that practitioners can come to the conclusion that a new framework provides the clearest and most comprehensive unifying perspective available. When the value judgment is effectively unanimous, as it is in the case of perfectoid geometry, it deserves to be considered as objective as the existence of the field itself.

[1] From What are the required backgrounds of Robin Hartshorne’s Algebraic Geometry book?:

Hartshorne’s book is an edulcorated version of Grothendieck and Dieudonné’s EGA, which changed algebraic geometry forever.
EGA was so notoriously difficult that essentially nobody outside of Grothendieck’s first circle (roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron.

Things began to change with the appearance of Mumford’s mimeographed notes in the 1960’s, the celebrated Red Book, which allowed the man in the street (well, at least the streets near Harvard ) to be introduced to scheme theory.

Then, in 1977, Hartshorne’s revolutionary textbook was published. With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.

However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.

There are many such books nowadays but my favourite is probably Basic Algebraic Geometry, volume 1 by Shafarevich, a great Russian geometer. …

The most elementary introduction to algebraic geometry is Miles Reid’s aptly named Undergraduate Algebraic Geometry, of which you can read the first chapter here.

Saharon Shelah, logic juggernaut

There’s a great quote via Hunter Johnson on Quora arguing for the importance of mathematical logic to math:

Shelah is attending a mathematics talk. The presenter has offered, with great difficulty, a new example of some mathematical structure, let’s say a quasi-Hebrand reticular matrixoid. The existence of a new example of this object is significant in the field of quasi-Hebrand reticulation theory (I am making up these names). 

Shelah has come in late and missed most of the talk. When the time for questions comes, he raises his hand and says, “I can give you uncountably many of these objects. Now, tell me, what is a quasi-Hebrand reticular matrixoid?”

Mathematical logic is about the forest rather than the trees. When you look at the structure that different mathematical fields have in common, you see overarching themes that make the theory work.

The funny thing is that it’s entirely believable, because this is Saharon Shelah we’re talking about. I don’t know of anyone else alive more prolific in math than he is – 1,166 papers/preprints/books as of July 2019 with 260 coauthors.

I was pretty happy to see him being signal-boosted today on Quora, by Alon Amit. Alon wrote:

He is also recognized as one of the most powerful problem solvers around (I remember this actual phrase being used in a Scientific American article about Van der Waerden’s Theorem). As a result, some mathematicians are slightly relieved that Shelah focuses on infinite combinatorics and model theory and not on their field.

Shelah made tremendous contributions to model theory and set theory, solving a huge number of open problems and establishing major theories and directions for research, most notably PCF theory. In the well-known classification of mathematicians into “theory builders” and “problem solvers”, Shelah is a rare dual citizen.

A “dual citizen” of the highest order! I’ve only heard this explicitly said (in terms of praise) about one other modern-day mathematician, Akshay Venkatesh in his Fields laudatio, but of course that’s just memory failing me.

What is Shelah’s style? A Kanamori gave a beautiful description back in 1999. Unfortunately it’s just one solid wall of text, too much for my non-Shelahian working memory, so I’ve broken it up:

In set theory Shelah is initially stimulated by specific problems. He typically makes a direct, frontal attack, bringing to bear extraordinary powers of concentration, a remarkable ability for sustained effort, an enormous arsenal of accumulated techniques, and a fine, quick memory.

When he is successful on the larger problems, it is often as if a resilient, broad-based edifice has been erected, the traditional serial constraints loosened in favour of a wide, fluid flow of ideas and the final result almost incidental to the larger structure. What has been achieved is more than just a succinctly stated theorem but rather the establishment of a whole network of robust constructions and arguments. A telling point is that when some local flaw is pointed out to Shelah, he is usually able to come up quickly with another idea for crossing that bridge.

Shelah’s written accounts have acquired a certain notoriety that in large part has to do with his insistence that his edifices be regarded as autonomous mental constructions. Their life is to be captured in the most general forms, and this entails the introduction of many parameters. Often, the network of arguments is articulated by complicated combinatorial principles and transient hypotheses, and the forward directions of the flow are rendered as elaborate transfinite inductions carrying along many side conditions. The ostensible goal of the construction, the succinctly stated result that is to encapsulate it in memory, is often lost in a swirl of conclusions. This can make for difficult and frustrating reading, with the usual problem of presenting a mathematical argument in linear form exacerbated by the emphasis on the primacy of the construction itself and its overarching generality.

Further difficulties ensue from the nature of the enterprise. Shelah regards the written word as necessary and central for capturing and fixing a construction, and so for him getting everything down on paper is of crucial importance. The tensions among the robustness of the construction, the variability of its possible renditions, and the need to convey it all in print are inevitably complicated by the speed with which he is able to establish new results. The papers have to be written quickly, previous constructions are newly refreshed and modified, and so a labyrinthian network may result over a series of related papers.

In mathematics one often aspires to the most elegant or definitive treatment; in contrast, Shelah’s work features a continuing, dynamic self-dialogue, one that pushes to the limits of exposition. Many may consider Shelah’s work to be “technical”, but as T S Eliot has written “We cannot say at what point ‘technique’ begins or where it ends” [‘The Sacred Wood’]. While there is a particular drive to solve specific problems, Shelah with his generalizing approach is able to draw out larger, recurring patterns that lead to new techniques that soon get elevated to methods.

What’s an example of this approach?

One primary instance is the whole complex of approaches and results he developed under the general rubric of proper forcing. Shelah started out in model theory, developing an abstract classification theory for models which is a continuing research program for him and model theorists to this day. In the mid-1970’s, in his first major body of results in set theory, Shelah resolved a long-standing problem in abelian group theory, Whitehead’s problem, by establishing both the consistency and the independence of the corresponding proposition. It is through these beginnings, motivated by the set-theoretic problems that arose, that Shelah started to develop a general theory of iterated forcing for the continuum.

But what does Shelah “do all day at the office”? In his own words:

Since I have succeeded in demonstrating a substantial number of theorems, I have also a lot of work completing and correcting the demos. As I write, I have a secretary typing (I did have a lot of troubles concerning this) and I have to proof-read a lot. I write and make corrections, send to the typist, get it back and revise it again and again.

A great amount of time is used to verify what I wrote. If it is not accurate or utterly wrong, I ask myself what went wrong. I tell myself: there must be a hole somewhere, so I try to ll it. Or perhaps there is a wrong way of looking at things or a mistake of understanding. Therefore one must correct or change or even throw everything and start all over again, or leave the whole matter. Many times what I wrote first was right, but the following steps were not, therefore one should check everything cautiously. Sometimes, what seems to be a tiny inaccuracy leads to the conclusion that the method is inadequate.

I have a primeval picture of my goal. Let us assume that I have heard of a problem and it seems alike to problems that I know how to resolve, provided we change some elements. It often happens that, having thought of a problem without solving it, I get a new idea. But if you only think or even talk but not sit down and write, you do not see all the defaults in your original idea.

Writing does not provide a 100% assurance, but it forces you to be precise. I write something and then I get stuck and I ask myself perhaps it might work in another direction? As if you were pulling the blanket to one part and then another part is exposed. You should see that all parts are integrated into some kind of completeness.

Indeed, sometimes you are happy in a moment of discovery. But then you find out, while checking up, that you were wrong. There has been a joy of discovery, but that is not enough, for you should write and check all the details. My office is full of drafts, which turned to be nonsense.

Strong words, but it does explain A Kanamori’s remark that “Shelah regards the written word as necessary and central for capturing and fixing a construction, and so for him getting everything down on paper is of crucial importance“.

Terry Tao: can an approach used to prove almost all cases be extended to prove all cases?

Recently Terry Tao posted to the arXiv his paper Almost all Collatz orbits attain almost bounded values, which caused quite the stir on social media. For instance, this Reddit post about it is only a day old and already has nearly a thousand upvotes; Twitter is abuzz with tweets like Tim Gowers’:

(this sentiment seems off coming from the editor of the Princeton Companion to Mathematics, a T-shaped mathematician with both bar and stem thick, not to mention a fellow Fields medalist)

The first comment on his post, by goingtoinfinity, voices the unasked question everyone’s wondering:

What is the relations between results for “almost all” cases vs. subsequent proofs of the full result, from historic examples? Are there good examples where the former influences the developments of the later? Or is it more common that, proving results for full results of a mathematical question, is conducted in an entirely different way usually?

As an example, Falting’s proof that there are only finitely many solutions to Fermat’s Last Theorem — did his techniques influence and appear in Wiles’s/Taylor’s final proof?

Terry’s response is the raison d’être of this post. It also features really long paragraphs, too long for my poor working memory, so I’ve broken it up for personal edification:

One can broadly divide arguments involving some parameter (with a non-empty range) into three types: “worst case analysis”, which establish some claim for all choices of parameters; “average case analysis”,which establish some claim for almost all choices of parameters; and “best case analysis”, which establish some claim for at least one choice of parameters.

(One can also introduce an often useful variant of the average case analysis by working with “a positive fraction” of choices rather than “almost all”, but let us ignore this variant for sake of this discussion.)

There are obvious implications between the three: worst case analysis results imply average case analysis results (these are often referred to as “deterministic arguments”), and average case analysis results imply best case analysis results (the “probabilistic method”). In the contrapositive, if a claim fails in the average case, then it will also fail in the worst case; and if it fails even in the best case, then it fails in the average case.

However, besides these obvious implications, one generally sees quite different methods used the three different types of results. In particular, average case analysis (such as the arguments discussed in this blog post) gets to exploit methods from probability (and related areas such as measure theory and ergodic theory); best case analysis relies a lot on explicit constructions to design the most favorable parameters for the problem; but worst case analysis is largely excluded from using any of these methods, except when there is some “invariance”, “dispersion”, “unique ergodicity”, “averaging” or “mixing” property that allows one to derive worst-case results from average-case results by showing that every worst-case counterexample must generate enough siblings that at they begin to be detected by the average-case analysis.

For instance, one can derive Vinogradov’s theorem (all large odd numbers are a sum of three primes) from a (suitably quantitative) almost all version of the even Goldbach conjecture (almost all even numbers are the sum of two primes), basically because a single counterexample to the former implies a lot of counterexamples to the latter (see Chapter 19.4 of Iwaniec-Kowalski for details).

At a more trivial (but still widely used) level, if there is so much invariance with respect to a parameter that the truth value of a given property does not actually depend on the choice of parameter, then the worst, average, and best case results are equivalent, so one can reduce the worst case to the best case (such arguments are generally described as “without loss of generality” reductions).

However, in the absence of such mixing properties, one usually cannot rigorously convert positive average case results to positive worst case results, and when the worst case result is eventually proved, it is often by a quite different set of techniques (as was done for instance with FLT). So it is often better to think of these different types of analysis as living in parallel, but somewhat disjoint, “worlds”.

(In additive combinatorics, there is a similar distinction made between the “100% world”, “99% world”, and “1% world”, corresponding roughly to worst case analysis and the two variants of average case analysis respectively, although in this setting there are some important non-trivial connections between these worlds.)

In the specific case of the Collatz conjecture, the only obvious invariance property is that coming from the Collatz map itself (N obeys the Collatz conjecture Col_min(N) = 1 if and only if Col(N) does), but this seems too weak of an invariance to hope to obtain worst case results from average case ones (unless the average case results were really, really, strong).