20160628 Peter Scholze Arithmetic Geometry Profile

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Quanta Magazine

https://www.quantamagazine.org/20160628-peter-scholze-arithmetic-geometry-profile/ June 28, 2016

The Oracle of Arithmetic

At 28, Peter Scholze is uncovering deep connections between number theory and geometry.

Nyani Quarmynefor Quanta Magazine

By Erica Klarreich

In 2010, a startling rumor filtered through the number theory community and reachedJared

Weinstein. Apparently, some graduate student at the University of Bonn in Germany had written a

paperthat redid Harris-Taylor a 288-page book dedicated to a single impenetrable proof in

number theory in only 37 pages. The 22-year-old student, Peter Scholze, had found a way to

sidestep one of the most complicated parts of the proof, which deals with a sweeping connection

between number theory and geometry.

It was just so stunning for someone so young to have done something so revolutionary, said

Weinstein, a 34-year-old number theorist now at Boston University. It was extremely humbling.

Mathematicians at the University of Bonn, who made Scholze a full professor just two years later,

were already aware of his extraordinary mathematical mind. After he posted his Harris-Taylor paper,

experts in number theory and geometry started to notice Scholze too.

Since that time, Scholze, now 28, has risen to eminence in the broader mathematics community.Prize citations have called him already one of the most influential mathematicians in the world and

a rare talent which only emerges every few decades. He is spoken of as a heavy favorite for the

Fields Medal, one of the highest honors in mathematics.

Read the related

Abstractions post:

Handicapping the

2018 Fields Medal

Scholzes key innovation a class of fractal structures he calls perfectoid

spaces is only a few years old, but it already has far-reaching ramifications

in the field of arithmetic geometry, where number theory and geometry come

together. Scholzes work has a prescient quality, Weinstein said. He can see

the developments before they even begin.

Many mathematicians react to Scholze with a mixture of awe and fear and

exhilaration, said Bhargav Bhatt, a mathematician at the University of

Michigan who has written joint papers with Scholze.
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Thats not because of his personality, which colleagues uniformly describe as grounded and

generous. He never makes you feel that hes, well, somehow so far above you, said Eugen

Hellmann, Scholzes colleague at the University of Bonn.

Instead, its because of his unnerving ability to see deep into the nature of mathematical

phenomena. Unlike many mathematicians, he often starts not with a particular problem he wants to

solve, but with some elusive concept that he wants to understand for its own sake. But then, said

Ana Caraiani, a number theorist at Princeton University who has collaborated with Scholze, the

structures he creates turn out to have applications in a million other directions that werent

predicted at the time, just because they were the right objects to think about.

Learning Arithmetic


The Mathematical Institute of the University of Bonn in Germany.

Scholze started teaching himself college-level mathematics at the age of 14, while attending

Heinrich Hertz Gymnasium, a Berlin high school specializing in mathematics and science. At

Heinrich Hertz, Scholze said, you were not being an outsider if you were interested in

mathematics.

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century

problem known as Fermats Last Theorem, which says that the equation xn+yn=znhas no nonzero

whole-number solutions if nis greater than two. Scholze was eager to study the proof, but quicklydiscovered that despite the problems 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, thats 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.

As Scholze burrowed into the proof,he became captivated by the mathematical objects involved

structures called modular formsand elliptic curvesthat mysteriously unify disparate areas of

number theory, algebra, geometry and analysis. Reading about the kinds of objects involved was

perhaps even more fascinating than the problem itself, he said.

Scholzes mathematical tastes were taking shape. Today, he still gravitates toward problems that

have their roots in basic equations about whole numbers. Those very tangible roots make even

esoteric mathematical structures feel concrete to him. Im interested in arithmetic, in the end, he

said. Hes happiest, he said, when his abstract constructions lead him back around to small

discoveries about ordinary whole numbers.

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. Notjust understand, but really understand on some kind of deep level, so that he also would not forget.

Scholze began doing research in the field of arithmetic geometry, which uses geometric tools to
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understand whole-number solutions to polynomial equations equations such as xy2+ 3y = 5 that

involve only numbers, variables and exponents. For some equations of this type, it is fruitful to study

whether they have solutions among alternative number systems called p-adic numbers, which, like

the real numbers, are built by filling in the gaps between whole numbers and fractions. But these

systems are based on a nonstandard notion of where the gaps lie, and which numbers are close to

each other: In a p-adic number system, two numbers are considered close not if the difference

between them is small, but if that difference is divisible many times by p.

Its a strange criterion, but a useful one. The 3-adic numbers, for example, provide a natural way to

study equations like x2= 3y

2, in which factors of three are key.

P-adic numbers are far removed from our everyday intuitions, Scholze said. Over the years,

though, they have come to feel natural to him. Now I find real numbers much, much more confusing

than p-adic numbers. Ive gotten so used to them that now real numbers feel very strange.

Mathematicians had noticed in the 1970s that many problems concerning p-adic numbers become

easier if you expand the p-adic numbers by creating an infinite tower of number systems in which

each one wraps around the one below it ptimes, with the p-adic numbers at the bottom of the tower.At the top of this infinite tower is the ultimate wraparound space a fractal object that is the

simplest example of the perfectoid spaces Scholze would later develop.

Scholze set himself the task of sorting out why this infinite wraparound construction makes so many

problems about p-adic numbers and polynomials easier. I was trying to understand the core of this

phenomenon, he said. There was no general formalism that could explain it.

He eventually realized that its possible to construct perfectoid spaces for a wide variety of

mathematical structures. These perfectoid spaces, he showed, make it possible to slide questions

about polynomials from the p-adic world into a different mathematical universe in which arithmetic

is much simpler (for instance, you dont have to carry when performing addition). The weirdest

property about perfectoid spaces is that they can magically move between the two number systems,

Weinstein said.

This insight allowed Scholze to prove part of a complicated statementabout the p-adic solutions to

polynomials, called the weight-monodromy conjecture, which became his 2012 doctoral thesis. The

thesis had such far-reaching implications that it was the topic of study groups all over the world,

Weinstein said.

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.


Peter Scholze in June at a geometry seminar at the University of Bonn.

Flying Over the Jungle

Despite the complexity of perfectoid spaces, Scholze is known for the clarity of his talks and papers.

I dont really understand anything until Peter explains it to me, Weinstein said.
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Scholze makes a point of trying to explain his ideas at a level that even beginning graduate students

can follow, Caraiani said. Theres this sense of openness and generosity in terms of ideas, she said.

And he doesnt just do that with a few senior people, but really, a lot of young people have access to

him. Scholzes 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 Scholzes 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 youre in the middle of the jungle and its actually very hard. But Scholze himself, Hellmann

said, would never lose himself in the jungle, because hes never trying to fight the jungle. Hes

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 cant do too complicated things.

While other mathematicians are now starting to grapple with perfectoid spaces, some of the most

far-reaching discoveries about them, not surprisingly, have come from Scholze and his collaborators.

In 2013, a result he posted online really kind of stunned the community, Weinstein said. We had

no idea that such a theorem was on the horizon.

Scholzes resultexpanded the scope of rules known as reciprocity laws, which govern the behavior

of polynomials that use the arithmetic of a clock (though not necessarily one with 12 hours). Clock

arithmetics (in which, for example, 8 + 5 = 1 if the clock has 12 hours) are the most natural and

widely studied finite number systems in mathematics.

Reciprocity laws are generalizations of the 200-year-old quadratic reciprocity law, a cornerstone of

number theory and one of Scholzes personal favorite theorems. The law states that given two prime

numbers pand q, in most cases pis a perfect square on a clock with q hours exactly when qis a

perfect square on a clock with p hours. For example, five is a perfect square on a clock with 11

hours, since 5 = 16 = 42, and 11 is a perfect square on a clock with five hours, since 11 = 1 = 1 2.

I find it very surprising, Scholze said. On the face of it, these two things seem to have nothing to

do with each other.

You can interpret a lot of modern algebraic number theory as just attempts to generalize this law,Weinstein said.

In the middle of the 20th century, mathematicians discovered an astonishing link between

reciprocity laws and what seemed like an entirely different subject: the hyperbolic geometry of

patterns such as M.C. Eschers famous angel-devil tilings of a disk. This link is a core part of the

Langlands program, a collection of interconnected conjectures and theorems about the

relationship between number theory, geometry and analysis. When these conjectures can be proved,

they are often enormously powerful: For instance, the proof of Fermats Last Theorem boiled down

to solving one small (but highly nontrivial) section of the Langlands program.

Mathematicians have gradually become aware that the Langlands program extends far beyond the

hyperbolic disk; it can also be studied in higher-dimensional hyperbolic spaces and a variety of other

contexts. Now, Scholze has shown how to extend the Langlands program to a wide range of
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structures in hyperbolic three-space a three-dimensional analogue of the hyperbolic disk and

beyond. By constructing a perfectoid version of hyperbolic three-space, Scholze has discovered an

entirely new suite of reciprocity laws.

Peters work has really completely transformed what can be done, what we have access to,

Caraiani said.

Scholzes result, Weinstein said, shows that the Langlands program is deeper than we thought its more systematic, its ever-present.

Fast Forward


Known for his work on perfectoid spaces, the 28-year-old Scholze has been called one of the most influential

mathematicians in the world.

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 doesnt know which does happen then, well, lucky you, because youve 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.

There was one occasion, though, when Scholze himself did hurry while trying to finish up a paper

in late 2013, shortly before the birth of his daughter. It was a good thing he pushed himself then, he

said. I didnt get much done afterwards.

Becoming a father has forced him to become more disciplined in how he uses his time, Scholze said.

But he doesnt have to make a point of blocking off time for research mathematics simply fills all

the spaces between his other obligations. Mathematics is my passion, I guess, he said. I always

want to think about it.

Yet he is not at all inclined to romanticize this passion. Asked if he felt he was meant to be a

mathematician, he demurred. That sounds too philosophical, he said.

A private person, he is somewhat uncomfortable with his growing celebrity (in March, for example,

he became the youngest recipient ever of Germanys prestigious Leibniz Prize, which awards 2.5

million euros to be used for future research). At times its a bit overwhelming, he said. I try to not

let my daily life get influenced by it.

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. Hechanged the way [the experts] think about it.

It can be scary but also exciting for other mathematicians when Scholze enters their field, Bhatt
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said. It means the subject is really going to move fast. Im ecstatic that hes working in an area

thats 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. Im still in the phase where Im trying to learn

whats there, and maybe rephrasing it in my own words, he said. I dont feel like Ive actually

started doing research.

20160628 Peter Scholze Arithmetic Geometry Profile

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