Mathematics: It’s About the Future
How some predictions fared in 2020 and other years
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Simon Donaldson, Maxim Kontsevich, Terence Tao, Richard Taylor, and Jacob Lurie (photo order) won the 2015 Breakthrough Prize in Mathematics. This did not happen since Thursday; it happened last June. When Tao was asked to explain the 2015 prize date at the start of his 11/12/14 appearance on The Colbert Report, he said,
“It’s about the future.”
Today Dick and I salute the prize-winners, and preview a new book about advances that were made in the years 2015–2019.
The prizes are awarded also in Fundamental Physics and Life Sciences. The physics prize was founded by Russian entrepreneur Yuri Milner and his wife Julia, who were joined by Sergey Brin and Anne Wojcicki, Mark Zuckerberg and Priscilla Chan, and Jack Ma and Catherine Zhang for the whole foundation. Each award is $3 million, over twice the emolument of a Nobel prize.
Prizes, in general, have been criticized as an inefficient way to reward past achievements. However, Tao’s impression was confirmed to us by a possibly misdirected e-mail: the prize is awarded for breakthroughs in the future. This includes Tao’s project mentioned at the end of his Colbert appearance about “whether water can spontaneously blow up.”
The E-Mail
One consequence of publishing books is getting more e-mails. Some come from Amazon and other retailers putting the books you’ve written high on lists of titles recommended for you. Others are requests from your publishers to review other books and proposals, which is only fair since we are grateful for several in-depth reviews of our early sketch of our quantum algorithms textbook as well as the short reviews on the jacket. Most interesting, however, is that sometimes you get copied on internal business where a general suggestion might be run by you.
I had just gotten off the phone with Dick one Friday afternoon last month as he was sending me a post draft when the expected ‘beep’ on my machine turned out to be from one of our publishers, with PDF not LaTeX attached. The subject line said,
- Breakthrough 2020 frontmatter
I froze for a minute wondering if this meant something was wrong with our book. Then came the ‘beep’ from Dick’s e-mail. The other e-mail stayed at the bottom, which meant it had been post-dated in the manner of various spams that evade my filter. More puzzling, the time shown in my Thunderbird window was 2 minutes before Dick’s mail. I had to do several clicks to get the full datestamp:
12/13/19 15:16
Really ’19’ not ’14’ as expected. I saw Dick was cc-ed like me, but before calling him back I read the short message body:
Greenlight GLL on attached proposal for their and reader reactions.
Perhaps we were not supposed to be copied yet, but Dick agreed the intent was clear, so we both clicked open the PDF. We are happy to bring its contents to you now.
The Book
The title page read, Breakthrough 2020: Visions of the Future. The next page included John Rainwater and Sam Parc as editors. We had to hunt to find Rainwater’s track record in functional analysis, but Parc was easy: she had numerous books including the beautiful 50 Visions of Mathematics, which had maybe influenced this title. The dedication page leapt out:
To all the co-workers whose efforts made the past five years a new Golden Age of mathematics.
Still thinking 2015, I—and Dick too—wondered how closely their selection would match the recent results we’d highlighted in 5+ years on this blog. But the Table of Contents brought nothing we’d seen before:
- Preface ………………………………………….. by Sergey Brin xi
- Acknowledgments …………………………. xv
- (No) More Secrets ………………………. by Maxim Kontsevich 1
- Exploding Universes ………………….. by Terence Tao 37
- Interfaces ……………………………………. by Simon Donaldson and Edward Witten 63
- ABC and the D Theorem …………….. by Richard Taylor and Shinichi Mochizuki 105
- Analyzing Stacks With Stacks ……. by Jacob Lurie 139
- The Supersymmetry Wall ………….. by Nima Arkani-Hamed and Leonard Susskind 187
- Shaking Off the Dust ………………….. by Alan Guth and Andrei Linde 205
The preface explained much but left other things hanging:
When we established the Breakthrough Prizes we did not intend to destroy the world economy, much less destroy the universe. We almost destroyed our companies, which were already affected by the 2015 Hacker War, but government bailouts from Amazon to us to Zynga helped erect the Quinternet to restore secure e-commerce. Although quantum networking and Qubitcoin-2 took millions of people on the outside, Maxim Kontsevich describes the inside in thirty-six pages. The theme of destruction and creation continues with Terry Tao. He began with a childlike question: “can water explode?” Of course it doesn’t, we think. But the entire universe is immersed in “water” called the Higgs Field, and from 2012 we know it is only semi-stable. This seemed horrible news, but Terry’s work hints it may be necessary to support informational processes that can generate life.
Simon Donaldson’s doctoral work showed the existence of 4-manifolds having topological but not differentiable correspondences to the standard one, but does Nature avail herself of them? His work with Ed Witten at the Stony Brook Simons Center during the Long Island Quarantine expands the possibility of yes. Geometry and numbers were joined in amazing claims by Shinichi Mochizuki in 2012, but it took Richard Taylor to close a hole in the proof of the “ABC Conjecture.” What they call “D-theory” for Dirichlet goes even further toward proving conjectures by Robert Langlands that once seemed a pipe dream. Jacob Lurie sprang open some questions in computational complexity by applying his higher algebra to analyze two-way pushdown automata. This does not yet solve P versus NP, but he shows how it stratifies the possible relationships between computational time and computational space.
The excerpt ended there. Nothing about the last two physics chapters. Clearly the contents too were incomplete. Perhaps they were considering life-science chapters or more physics but needed more editors? The first seven chapters seemed to be finished, given the page numbers, but were not attached. What could they contain?
Our Guesses on the Book
The physics chapters left out of the preface were the easiest for me and Dick to guess. Arkani-Hamed and Susskind must have done something—or would do something—to explain why supersymmetry could not be observed by the Large Hadron Collider. Perhaps the pervasive spatial tension accompanying a positive cosmological constant simply walls off creating even an echo of conditions under supersymmetry. From Guth’s and Linde’s title, evidently the flaws in last year’s analysis of B-mode polarization in the cosmic microwave background had been fixed with higher confidence than now, confirming gravity waves and inflation.
Kontsevich’s title obviously referred to the 1992 Robert Redford movie “Sneakers,” whose premise is that factoring has been efficiently solved—classically. Perhaps Kontsevich’s approximations to Richard Feynman’s path integrals broke the parts of Peter Shor’s quantum factoring algorithm that had not already been shown classically approximable. Dick and I could not tell what “D-theory” is supposed to be; we thought of “D-maps” for the Jacobian conjecture but those are named for someone else. We already riffed on Tao’s computational Navier-Stokes theory in our 2014 April Fool’s post, but apparently it was no joke.
I recognized the pun in Lurie’s title, since while writing our memorial post on Alexander Grothendieck I had perused his 600+ page manuscript “Pursuing Stacks.” A paper by Lurie and Dennis Gaitsgory which Bill Gasarch cited as “solving a real problem” runs to 394 single-spaced wide journal pages, so I marveled at the chapter staying under 50 pages. Howard Straubing’s book Finite Automata, Formal Logic, and Circuit Complexity laid out connections from finite automata to category theory and algebra, while connections between pushdowns and complexity were shown 40 years ago by Zvi Galil, so I guess connecting pushdown automata to category theory and algebra was a difficult but possible next step.
The Donaldson-Witten title harked me back to a dinner thirty years ago at Merton College, Oxford, in honor of Donaldson. I was then a Junior Fellow of the college, and was seated across from Witten at the High Table. Just as the main course was served, I asked Witten whether it was possible for two of Donaldson’s manifolds to be joined so that an arbitrarily large distance in one could be traveled by crossing, going a short distance in the other, and crossing back. To my surprise this was not a silly question; here is the same idea for a simple wormhole, but I had higher-dimensional gluing in mind. Witten rhapsodized in reply for over 30 minutes without once touching his food. I ate gingerly trying not to interrupt but eventually cleared my plate, as did everyone else, while the hubbub of the lower student tables subsided to silence as the great hall emptied. I recall the serving staff standing helplessly by, since protocol prevented the next course from starting until everyone had finished, with their gazes fixed on the unbroken Cornish hen or similar bird. Possibly after an offer to substitute a warm one, Witten finally ate with due dignity while the rest of us discussed various subjects, before all progressed equally to the next courses.
2014 Predictions Scorecard
Although our last year’s predictions were jocular, we will still score them. Perhaps all “predictions” should be taken no more seriously than those on last Thursday’s Rose Bowl outcome.
Two integers of over one million digits each, in decimal, will be discovered so that
Wrong: Okay we were kidding, but it would have been fun—no?
Trading in Bitcoins will be stopped when an anonymous team posts an algorithm that breaks the scheme. Social media will be abuzz with the question: why did they post it? They could have made billions in real dollars.
Wrong: Okay but we still bet they would have gone for the fame. We should have predicted instead that a Bitcoin company would sponsor a college bowl game.
A new field of computer science called computational football will be one of the hottest areas of research.
Half-Wrong: Since ESPN pays almost a billion US dollars per year for rights to college football, something computational must be going on.
A proof that there is a proof that there is a proof that there is no proof that there is a proof that there is a proof in Peano Arithmetic of will be found.
Wrong, or don’t we know?
Computer scientists will sweep the all the Nobel Prizes except Peace and Literature.
Wrong. We might have been right about the Literature prize though, since its unexpected winner was once tutored in geometry by the computational novelist Raymond Queneau.
A company called Braincloud LLC will announce a competitor for Google Glass that is controlled directly by brain pulses.
Wrong. Though possibly things like it are being surreptitiously tested at chess.
A new massively-multiplayer online role-playing computer game called “DoS Survivor” will take the world by storm.
Pretty close: the massive Xbox and PlayStation denial-of-service hack on Christmas Day made players of many.
MOOCs will adopt a “human-centered support structure.” This will involve geographically localized cells of up-close instruction at regularly spaced time intervals, with generous time in-between for absorbing material and practicing non-bubble exercises.
Right: They are called classrooms. Yes.
Three reader predictions in comments to last year’s post were:
- “A computer will make an amazing conjecture that looks true but nobody can prove—not even a computer.”
- “Schemes for quantum correction algorithms will begin being viewed as how to restrict the free-will of systems.”
- “Someone from very rich people (Bill Gates or Sergey Brin, for example) [will] make Nobel-2 Prize (like Nobel Prize) for CS.”
These seem to have come closer than most of ours. The CIA and the movie “Back to the Future II” also came closer. For best result of 2014 we nominate discoveries by Tao and others about gaps between primes.
Open Problems
Our book chapters amount to predictions. How will they fare by 2020? You are welcome to put your own predictions in the comments.
Happy New Year from GLL.
[word changes—see first KWR comment]
In the Witten dinner story, I changed “Possibly their offer” to “Possibly an offer” to make clearer that I don’t recall such an offer, only some less-definite memory of a motion from the staff. I changed “final piece” before “of Shor’s algorithm” to just “part”—it’s really in the interface between the quantum trials and the classical use of them. I considered titling ch. 6 “Supersymmetry Equals Death” to catch the interpretation I glean from Susskind’s book The Cosmic Landscape, but stayed more topical—still keeping Susskind.
Celebritism fosters tolerance for plutocracy. That’s why those prone to the Dark Side handsomely finance it, if wealthy enough.
Recently revealed documents exposed massive tax evasion by the very corporations of these individuals who set-up these prizes. Through Luxembourg alone, more than 100 billion dollars of “legal” tax evasion, from American companies alone. Each year. That’s nearly 10% of the entire Federal budget of the USA.
The whistle blower, a computer specialist, was indicted. The tax evader in chief, J-C Junker, was made head of the European Commission. Meanwhile, basic science is getting squeezed.
Crumbs offered to a few thinkers do not replace the diminution of public financing for the most advanced research, and basic education. They are part of a cover-up hiding the fact that austerity is for everybody except for plutocrats and their extravagance. It is part of a magic show: as the plutocrats brandish their glittering prizes, they divert everybody’s attention from their massive tax evasion, and how they have captured the public discourse to foster their ever increasing power.
This is all part of the Great America!
2015’s biggest news arrived in January, with the announcement by David Mumford and John Tate (in the 29 January 2015 issue of Nature) of the recent completion of Alexander Grothendieck’s program to systematically rewrite all ten volumes of the Landau and Lifshitz Course on Theoretical Physics, as a unified narrative that respects — openly, comprehensively, and scrupulously — 21st century ideals of logical universality, mathematical naturality, and informatic physicality.
Like the Niagara-spanning kite-string of Homer Walsh, these fascicles — lovingly edited by Donald Knuth, Colin McLarty, and Jacques Roubaud — carry readers all the way from Grothendieck’s starting question “What is a metre?” to Laura Shigihara’s and Emmy Toyonaga’s open-ended closing question “How does healing work exactly?” (Rakuen, 2015).
Texts that include Quantum Algorithms Via Linear Algebra (Lipton and Regan, 2014) and The Singular Universe and the Reality of Time (Unger and Smolin, 2014) provide the needed linkage.
In accord with Grothendieck’s guiding vision, namely, that pursuing the question “what is a metre?” serve all living people, the release of these fascicles has been coordinated by the member-nations of the BIPM, under the aegis of the Convention du Mètre (1875).
—————
Readings
I expect something like this will happen in 2015:Microbes take over and then destroy the HAL 9000 prototype.
Overtaken by events: F-35 delayed after fourth prototype becomes self-aware (2014). Fortunately the STEAM community will be OK … so long as graduate students don’t become self-aware!
That’s quite the opposite prediction John Sidles, from your link the F35 has to be put down because it attains sentience. In my prediction HAL9000 does not last more than 9s after being exposed to alife. Sentience does not trump microbes, not even in the meat space.
The intended point is that cognition without compassion is inviable. There is a large, sophisticated (and even funny) body of literature to this effect … however large cohorts of the STEAM community are unfamiliar with this literature, and/or find great difficulty in comprehending it, and/or dismiss it prejudicially.
Fortunately, fresh new channels of understanding emerging. Works like the Shigihara-Toyonaga-Holmberg game Rakuen — which will be released later this month — can be experienced as 21st century embodiments of Donald Knuth’s celebrated dictum:
Questions What emerging Knuthian science are computer-works like Rakuen (2015) — and its predecessor To the Moon (2011) — presenting to us? What new elements are accreting to the STEAM-centric Knuthian art of works like Yannick Grannec’s The Goddess of Small Victories (2014)?
We can wonder too: how will these new elements be realized in mathematical theorems, physical understanding, innovative technologies, and new enterprises?
Audience These are questions that every aspiring STEAM-student — and every parent/spouse/child/mentor of a STEAM-student too — can pursue with benefit.
Aside The notion that humor can serve as an immune system for cognition — by neutralizing toxic “microbial” ideas — goes back at least as far as Marvin Minsky’s Society of Mind (1988, see in particular chapters “27.5 Jokes” and “27.6 Humor and censorship”).
Readings
“A proof that there is a proof that there is a proof that there is no proof that there is a proof that there is a proof in Peano Arithmetic of {\mathsf{P} \neq \mathsf{NP}} will be found.
Wrong, or don’t we know?”
—–
In other words, a proof leading to a formal contradiction derived from a premise that is true! This would be like finding one-way permutations in physical world. This proof may already exist.
What’s the point of repeatedly recognizing and congradulating the leaders of a field who are often already well-known and well-recognised in their communities mostly thanks to the impact of their contributions.These researchers often enjoy the financial support and the instutional privilages of their (often ivy-league) univeristies. If we believe in a more equal society, perhpas encouraging the less well-known researchers with prestitious prizes–who have not yet “made it” through their results–might have its fair share of societal benefits.
Pax,
I wholeheartedly agree with this.
Most people are not aware of the reasons behind the small minority (0.1% perhaps) which controls the US, and perhaps the whole world with their ever growing wealth and power.
Wealth, power arrogance, and injustice, they all go hand in hand.
The scientific system encourages a strong form of elitism. As Jean Dieudonné would say in his harsh, cut and dry fashion: “Good mathematics is made by few people. There are a handful of leaders. The right directions are those given by these people. The opinion of others is irrelevant. Those who follow are by no means negligible – they act as sounding boards.”
In regard to STEAM elitism …
The opposite lesson Jean Dieudonné’s words is communicated by his deeds, as recounted in the novel (or is it an autobiography?) by Jacques Roubaud, titled Mathématique: (1997)
Note Roubaud’s works are not easy reading. GLL readers who prefer (what amounts to) a succinct and actionable summary of Roubaud’s views (that is reasonably accurate, as it seems to me) can attend to Roberto Unger’s video essay (beginning at minute 8:51) at the link below.
GLL readers who are interested in feminist STEAM-contexts are invited to reflect upon the sobering fates of mathematician Marcelle Espiand (Sections 30 and 44-45) and of Roubaud’s philosopher-spouse Alix Cleo (whose presence is pervasive in Roubaud’s writings).
Ah yes, thank you John for reminding me of this delightful book by Jacques Roubaud. It’s a piece of choice in the French history of mathematics, at the peak of the Bourbaki era. I had read the original version when it was published in 1997. I enjoyed its literary, post-modernist style. Like you mention, it was translated into English in 2012 – though I wonder if such things should ever get translated… but well, I’m biased of course.
By the way, I’m not so sure Dieudonné’s deeds were in total contradiction with his words. Rather, he seems to have applied them to himself – and indeed, Grothendieck was very lucky to benefit from such a luxury sounding board! 😉
Serge, it is a pleasure to meet (on-line) a collegial Roubaud-fan. It was Lars Onsager who said:
My wife Constance is writing (slowly) a book upon the eagle-itarian topic “hope sustained” … my supportive role being to “munch the humus” of works like Roubaud’s and (more broadly) the STEAM-related ouvre of the OuLiPo and the Ou”X”Po.
In regard to which, the attention of Gödel’s Lost Letter readers is directed toward (chemical engineer and OuLiPo founder) Francois Le Lionnais’ Painting at Dora, which reads naturally as an essay upon “hope sustained”; hope that is sustained not by painting only, but by the entire STEAM enterprise
A beautifully printed edition of Lionnais’ short essay, as translated by OuLiPo member Daniel Levin Becker, has been issued recently by \text{roteotypes}$ press (url below) … it makes a thoughtful/hopeful gift!
John, you just gave me the ultimate evidence I needed for regarding you as the genuine reincarnation of Jean Dieudonné! 🙂
Le Lionnais was also a great chess specialist – in fact he was a specialist of many things. He also wrote an impressive “Dictionary of Mathematics”, a book on “Remarkable numbers”, and what I consider to be his masterpiece: the compendium “Great Trends in Mathematical Thought” – mostly written during WWII but published slightly afterwards – with contributions by Borel, Denjoy, Weil, Dieudonné, Bourbaki… The Dictionnary was the result of his personal collection of mathematical flashcards, while the two others were conceived during his imprisonment time in the Dora camp.