In the middle of December I was approached by an editor at New Scientist to write an article about “the work of Peter Scholze and its connections to the Langlands program, quantum theory, and anything else it might reasonably be said to have connections to.” Since the publication of my book, various people have been encouraging me to devote some time to writing popular accounts of the contents of mathematics, including contemporary (“cutting-edge”) work, and not just what my book calls “the mathematical life.” Scholze’s work is certainly cutting-edge, and I had already published a semi-philosophical account of his “perfectoid concept,” but the material seemed rather remote from what I imagined to be the concerns of the typical reader of New Scientist. The editor naturally mentioned the rumor that Scholze would be receiving a Fields Medal at next month’s International Congress of Mathematicians in Brazil, but for reasons that were not clear to me he seemed to feel that Scholze’s work would somehow have more resonance for his readers than that of the other potential laureates. Nevertheless, I accepted the challenge, and on February 1 I sent the editor a draft containing about 2/3 of the requested 2400 words, asking “whether the tone is appropriate and whether the mathematics is at the right level.”
Over the next two months there ensued the kind of lively give-and-take with the editor that I have always imagined to be the privilege of those who eke out their livings writing for the more intellectually ambitious of the mass-circulation magazines (Google tells me that New Scientist’s circulation in 2016 was 124,623). The editor wrote back the very same day to warn me that mathematics articles are typically a hard sell for a magazine like his, but that “with the right approach” they can be successful. It would be important for me to convince readers — at the very least, those “who might know something about Fermat’s Last Theorem or the Riemann Hypothesis” — that they should care about the material.
Taking these suggestions to heart, I sent the editor three more drafts, and by the middle of March I was ready to see how it would be transformed by the process, mysterious to me, known as “editing.” The result, when it arrived on March 28, was deeply discouraging. Very little of my own text had survived the cuts. In its place was an admittedly smoothly flowing narrative composed largely of the kinds of hackneyed metaphors and extraneous historical anecdotes that did nothing to clarify the originality of Scholze’s insight. After rapidly exchanging a few polite messages, the editor and I agreed that it was pointless to continue, and that it would be best if the New Scientist could salvage what it could from our correspondence and my previous draft; the editor promised to “run [these extracts] past [me] for approval before use.”
Six weeks passed, and since I had heard nothing from the editor I assumed the article had been “killed” (an expression I’ve already encountered in my interactions with journalists). But I checked during a lull in the middle of a lecture in Paris and was surprised to find that the New Scientist had gone ahead without notifying me and had published an article — a cover story! — under the Oscar-worthy title “The Shape of Numbers” (or the title “‘Perfectoid geometry’ may be the secret that links numbers and shapes”; or even “Theorem of everything: The secret that links numbers and shapes,” depending how you find it on the internet).
I’m not particularly happy that the author failed to let me know just how I was being quoted, and I don’t expect I’ll have anything to do with New Scientist in the future. And I don’t think it’s very helpful to have described Aristotle as an “ancient Greek philosopher and mathematician.” Still, even though the article doesn’t make much headway in explaining Scholze’s “secret that links numbers and shapes,” it could certainly have been worse.
The author preserved enough words from my final draft to render the draft unpublishable in any form, but I do believe I have the right to reproduce it on this blog. Please be indulgent when reading it and bear in mind that it is still just a draft, written for the eyes of the sympathetic and professional editor who still exists, if only in my imagination.
Number theory and geometry, the two most ancient branches of mathematics, could hardly be more different, at least on the surface. The former deals with the properties of integers — 1, 2, 3, and so on — and is designed to understand discrete objects. The latter studies spatial relations and measurements, and is built on our intuition of continuity. Aristotle thought they were separate because they applied to such distinct domains: “we cannot… prove geometrical truths by arithmetic,” he wrote, and he meant “and vice versa” as well.
Yet mathematicians have long speculated that features shared by arithmetic and geometry have common origins. The French mathematician André Weil described this to his sister in particularly vivid terms:
around 1820, mathematicians … permitted themselves, with anxiety and delight, to be guided by the analogy [between an arithmetical and a geometric theory]. [Now] gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur…
The unusual erotic charge of this letter, written in 1940, was stimulated by Weil’s pleasure in his recent solution of a geometric analogue of what then, as now, was the outstanding problem in number theory: the Riemann hypothesis. Like many problems in number theory, this one focuses on prime numbers, like 2, 3, 5: a number is prime if it can’t be factored as the product of two smaller numbers (unlike, say 6 = 2 x 3). There are infinitely many prime numbers, scattered among the integers according to no determinate pattern, but their frequency can be measured. The Riemann hypothesis predicts that this frequency follows the most natural possible rule.
The geometric version proved by Weil is the corresponding prediction for the frequency of points on a certain kind of curve. Just as prime numbers can be ordered by size, these points can be ordered by degree. Weil’s proof, which marks the beginning of the science with the most unaristotelian name of arithmetic geometry, showed that the number of points up to a given degree fits the prediction of the geometric Riemann hypothesis.
In a modern version of the analogy Weil found so delicious, prime numbers are points on a highly implausible kind of curve called Spec(Z), all stuck together by a strange sticky point that represents the familiar arithmetic of fractions. Ever since Weil proved his theorem about curves, and with increasing insistency in the last two decades, number theorists have believed that if one could make Spec(Z) genuinely curvy then fantastic consequences would follow — possibly including the Riemann hypothesis. Peter Scholze, today’s 30-year-old crown prince of arithmetic geometry, has not gone quite that far, but the p-adic geometry he has developed over the past 7 years has provided tantalizing hints of how a geometry of Spec(Z) might be built. In the process he has been transforming number theory at a rate that has the rest of us struggling to keep up.
Scholze, born in the former East Germany, would undoubtedly win a contest for World’s Most Popular Mathematician if there were such a thing; he has already received a long list of more conventional prizes. The Fields Medal is the highest honor for mathematicians under 40; most mathematicians are convinced that Scholze will be one of the winners at next August’s International Congress of Mathematicians in Rio de Janeiro. Scholze chose “p-adic geometry,” naturally enough, for the title of his prestigious plenary lecture at the Rio meeting. The “p” in “p-adic” denotes a prime number. Each prime has its own system of p-adic numbers, in which numbers become closer as their difference grows more divisible by p. In the 5-adic numbers, for example, 50 is 25 times closer to 2,000,000 than it is to 51 or 52. The 2-adic numbers are like binary numbers, but written in the wrong direction: our 16 is represented as 10000 in binary but is more like .0001 in the 2-adics.
There is an intrinsic geometry to the p-adic numbers, but it has little in common with Euclidean geometry. A p-adic circle would be composed of infinitely many smaller circles, in a fractal pattern, while all p-adic triangles would be isoceles. But you can’t actually draw p-adic circles or triangles — in fact, you can’t connect any two p-adic dots by anything resembling a straight line. P-adics were introduced by Kurt Hensel in 1897 as a way of understanding solutions to diophantine equations — polynomial equations with whole number coefficients. Perhaps the most famous diophantine equations are the Fermat equations
Xn + Yn = Zn
where the exponent n is a positive integer. When Sir Andrew Wiles proved in the early 1990s that the Fermat equation has no solutions when n >2 — this is the famous Fermat’s Last Theorem — practically every step in the proof involved p-adic numbers. Hensel’s version of p-adic geometry was barely relevant to Wiles’s work.
Scholze takes a different approach to p-adic geometry, taking his cue from the radical expansion of geometry in the 1960s under the leadership of Alexander Grothendieck. In contrast to the system inherited from Euclid, which dissected circles and triangles as singular objects, or the analytic geometry of Descartes, which studied parabolas and ellipses as if they were drawn on graph paper, each of Grothendieck’s geometric objects is at all times considered in relation to every other object in its category — the technical term for the principles contemporary mathematicians use to organize objects of a given type. So where a point in the Euclidean or Cartesian plane is just a familiar dot on a flat surface, a Grothendieck point is more like a way of thinking about the plane — which includes the possibility of drawing a triangle or an ellipse, or even squashing the surface of the globe into a planar map.
Grothendieck is usually considered the most influential mathematician of his time; the solution of Fermat’s Last Theorem, like every other major development in number theory over the past half century, would have been impossible without his innovations. Nevertheless the old Cartesian intuition, corrected by habits from algebraic calculation, largely sufficed when Grothendieck’s ideas were applied, notably by the French mathematician Jean-Marc Fontaine, who invented a series of new algebraic systems to bridge the gap between p-adic arithmetic and Grothendieck geometry. Scholze’s spaces, which retain some properties from familiar geometry and sacrifice some others, severely strain this intuition.
P-adic geometry can be viewed as the study of the geometry — in Grothendieck’s relational sense — that you would see hanging off the sticky curve Spec(Z) if you examined it under a microscope near the prime p. Scholze was only 24 when his dissertation introduced the theory of perfectoid spaces, which combined the best properties of the many kinds of Grothendieck-style p-adic geometries that had been studied over the previous half century with Fontaine’s p-adic number theory. In the intervening years Scholze and his collaborators have used perfectoid geometry to solve or clarify so many outstanding problems in number theory and in other branches of arithmetic geometry that last year’s annual Arizona Winter School on perfectoid spaces attracted a record 400 graduate students and postdocs — double the previous record.
Perfectoid geometry is very much a work in progress, and its details are dispersed among hundreds of pages of difficult mathematics, but one can begin to see the point with the help of Weil’s “slightly adulterous” analogy between algebra and geometry, as applied to differential calculus. As developed by Newton and Leibniz, calculus permits the application of the notions of geometry on an infinitesimal scale, predicting the motion of a particle under the influence of external forces. In the most familiar cases, this motion can be described as a function of time t by a Taylor series. This is an infinite version of a polynomial function of t:
f(t) = ∑ antn
where the coefficients an are constant real numbers. A p-adic number has a similar expression:
where the the coefficients an are now integers, but the variable t has been replaced by the prime number p. The two expressions have a completely different character, however: whereas t is a variable, and can therefore take on infinitely many values and trace a geometric figure as time varies, the number p is itself a constant and the p-adic expression belongs to pure algebra.
The aim of perfectoid geometry, in a single sentence, is to make the constant p behave like a variable, and thus to apply geometric methods to the arithmetic of p-adic numbers, and from there to the rest of number theory. This has a most disconcerting implication. Just as there are functions in calculus that depend on many variables — the forces on a vibrating string, for example, depend on the position along the string as well as time — perfectoid geometry makes it possible to clone a prime number, so that there can be several perfectoid versions of 3, taking independent values. With his theory of diamonds, a subsequent development of perfectoid spaces, Scholze managed precisely this.
Weil used a similar principle to prove his geometric Riemann hypothesis, which also depends on a prime number p. One can think of the curves he studied as trajectories of a particle parametrized by a time variable t. With a second variable u one can trace a second copy of the curve — a second particle—and Weil’s analysis of the frequency of points is based on using both copies simultaneously and comparing the places where the two particles coincide — this is the equation t = u — and where they have a fixed degree — this is given by a second equation (for example t = up means the point has degree 1). Similarly, in Scholze’s p-adic diamonds — take the prime p = 3 for concreteness —the 3-adic numbers stretch out into a kind of curve, and the excitement happens when my 3 gets close to colliding with your 3.
Apart from providing an especially rich framework for p-adic geometry, the most immediate applications of Scholze’s perfectoid spaces may be to the vast program outlined 50 years ago by Robert P. Langlands to unify number theory with the geometry of Lie groups, the systems of symmetries that are also central to mathematical physics. Mathematicians are aware that Wiles proved Fermat’s Last Theorem by establishing one particular consequence of the Langlands program; the last step was completed in collaboration with Richard Taylor. Scholze recently joined forces with Taylor and eight other mathematicians to push the argument of Wiles and Taylor in a new direction, one that would have been inaccessible without perfectoid spaces.
The full Langlands program is no more likely than the original Riemann hypothesis to be settled in the near future. But it also has a purely p-adic chapter. Scholze’s first published papers, before he invented perfectoid geometry, introduced a new perspective on this local Langlands correspondence — a subject on which I worked with Taylor about 20 years ago. More recently, the French mathematician Laurent Fargues proposed a way to use the cloning property of Scholze’s diamonds to provide a full solution of the p-adic side of the Langlands program. There are persistent rumors that Fargues and Scholze are working intensively on this proposal in advance of the coming summer’s meeting in Rio.
Scholze was briefly in the news in 2015 when he refused a $100,000 New Horizons Prize — the junior version of the $3 million Breakthrough Prizes awarded every year in a Hollywood-style extravaganza in Silicon Valley. Since he did not intend his decision as a public statement, guesses about Scholze’s motivations continue on the internet. What I can say is that parallels with the actions of Grigory Perelman, who solved the most famous problem in (traditional) geometry but refused the Fields Medal as well as the $1 million Clay Millenium Prize, before withdrawing from mathematics entirely, are completely off base. Perelman was portrayed in Masha Gessen’s Perfect Rigor as a hermit and a crank, with rigid ideas of what is and is not proper. Scholze is gregarious, thoughtful, generous with his ideas, actively supportive of junior colleagues (some of whom are slightly older than he is). He doesn’t seek publicity, and he is most likely to be spotted at a conference drinking beer with his friends, but he doesn’t mind talking to the press when necessary. In every way he has shown that he is ready to accept the responsibilities that the mathematical community generally expects of its most influential and respected individuals. My guess — but it’s no better than anyone else’s — is that he decided that the priorities of Silicon Valley are just not compatible with those of the mathematical community, as he sees it.
Whatever his reasons, mathematics needs more individuals like Peter Scholze. While the secrets of his success are not likely to be transmitted even to those who work most closely with him — and there are no prospects of cloning him in the near future — he has provided some insight into his goals as a mathematician, in a recent message that he has allowed me to share.
“What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.
Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.”