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.
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.
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.