Mathematics: It’s About the Future
How some predictions fared in 2020 and other years
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.”
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:
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 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
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.
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]