Arithmetic Oracle

Original author: Erica Klarreich
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At 28, Peter Scholze reveals the deep connections between number theory and geometry.




In 2010, the community of people who study the theory of numbers passed amazing hearing - and reached the Weinstein Jared [Jared Weinstein]. Allegedly, a graduate student from the University of Bonn in Germany published a work in which a 288-page proof of a theorem from number theory is reduced to just 37 pages. 22-year-old student Peter Scholze found a way to get around one of the most difficult parts of the proof by comparing number theory and geometry.

“It's just unbelievable that such a young man could do something so revolutionary,” says Weinstein, a 34-year-old number theory specialist at Boston University. “This is an undoubted reason for respect.”

The mathematicians at the University of Bonn, who awarded Scholz the title of professor just two years later, already knew about his extraordinary mental abilities. After the publication of the work, experts in both number theory and geometry began to notice it.

From that moment, Scholze, now 28, has grown to a high position already in the wider mathematical community. He is called " one of the most influential mathematicians in the world ", and "a rare talent that appears every few decades ." They talk about him as a favorite among applicants for the Fields Prize, one of the highest awards for a mathematician.

Scholze’s key innovation, the class of fractal structures called perfectoid spaces, was only a few years old, but it already leads to far-reaching consequences in the field of arithmetic geometry, in which number theory and geometry merge. Weinstein says Scholze's work was visionary. "He was able to see the consequences before they began to happen."

Bhargav Bhatt , a mathematician at the University of Michigan who wrote collaborations with Scholze, says many mathematicians react to his work "with a mixture of reverence, fear, and excitement."

And this is not because of his character, which colleagues describe as mundane and generous. "He never makes it clear to you that superior to you" - says Eugene Hellman [Eugen Hellmann], Scholze colleague at the university. Rather, it is because of its frightening ability to look so deep into the essence of a mathematical problem. Unlike many mathematicians, he begins work not with a specific problem requiring solution, but with some elusive concept that he wants to understand for the sake of interest. But then, according to Anna Karayani [Ana Caraiani], a Princeton University number theory specialist who worked with Scholze, the constructions he creates “discover applications in a million other directions that were not originally predictable - simply because the right objects were chosen for the study.”

Learn arithmetic



The Mathematical Institute at the University of Bonn, Germany

Scholze began to independently study institute mathematics at the age of 14, attending the Heinrich Hertz Grammar School, a Berlin school with a bias in mathematics and science. In this gymnasium, as Scholze described, "you were not a stranger if you were interested in mathematics."

At age 16, Scholze learned that ten years before that, Andrew Wiles had proved the famous 17th-century theorem known as Fermat’s Great Theorem , which states that the equation x n + y n = z nthere are no solutions in integers greater than zero for n> 2. Scholz really wanted to study the proof, but it quickly became clear that, despite the simplicity of the theorem, its proof uses mathematics of the most advanced level. “I didn’t understand anything, but it was very cool,” he says.

And Scholze began to study what knowledge gaps he needed to fill in order to understand this proof. “And until now, I usually teach everything like that,” he says. “I never learned basic things like linear algebra - I comprehended them by learning something else.”

Burrowing into the proof, he was struck by mathematical objects called modular forms and elliptic curves.that mysteriously combine disparate areas such as number theory, algebra, geometry, and analysis. According to him, the study of the types of objects used in the proof was perhaps even more interesting than the proof itself.

Scholze's mathematical tastes began to be determined. Today, he still gravitates to problems where simple equations and integers are found. And these tangible roots quite clearly allow him to feel even esoteric mathematical structures. “Essentially, I'm into arithmetic,” he says. According to him, he is most fortunate when his abstract constructions lead him back to small discoveries related to ordinary integers.

Upon graduation, Scholze continued to study number theory and geometry at the University of Bonn. As his classmate Helman recalls, Scholze did not write anything in his math classes. Helman argues that Scholze understood the course material in real time. “I didn’t just understand, but I understood at some deep level, which allowed him not to forget the material.”

Scholze began to study arithmetic geometry, using geometric tools to understand integer solutions of polynomial equations - such as xy 2+ 3y = 5, where only numbers, variables and degree are involved. For some of these equations, it is useful to find out if they have solutions in an alternative system of numbers called p-adic numbers. Like real numbers, they are constructed by filling in the voids between integers and fractions. But this system is based on the non-standard idea of ​​the location of these voids and the proximity of numbers to each other. In a p-adic system, two numbers stand close not when the difference between them is small, but when the difference between them is divided by the degree p (the larger the degree, the closer the numbers).

The criterion is strange but useful. For example, 3-adic numbers help more naturally study equations of the type x 2 = 3y 2 , in which the factor of three is key.

R-adic numbers "are far removed from everyday intuition," says Scholze. But over the years they became natural for him. “Now for me real numbers are more complicated than p-adic ones. I’m so used to them that the material ones seem much stranger to me. ”

In the 1970s, mathematicians noticed that many problems about p-adic numbers become easier if you expand these numbers with an infinite tower of numerical systems, in which each one wraps around the bottom p times, and p-adic numbers are at the bottom of this tower. At the “top” of the endless tower is a wrapping space - a fractal object, which is the simplest example of perfectoids that Scholze will later develop.

Scholze set himself the task of figuring out why these infinite wrapping constructions greatly simplify many problems associated with p-adic numbers and polynomials. “I tried to understand the essence of this phenomenon,” he says. “There was no single formalism that could explain it.”

At some point, he realized that it was possible to create perfectoid spaces for a wide variety of mathematical structures. He showed that these spaces make it possible to transfer questions related to polynomials from the world of p-adic numbers to other mathematical fields, where arithmetic is greatly simplified (for example, you do not need to carry over the transfer when adding). “The strangest property of perfectoid spaces is that they can magically move between two number systems,” says Weinstein.

Awareness of this allowed Scholz to prove part of a complex statement about p-adic solutions to polynomials, called the “weighted monodromy hypothesis”, and he designed it as a doctoral dissertation in 2012. “This work has such far-reaching implications that it has become a subject of study for groups of scientists around the world,” says Weinstein.

Helman says that Scholze “found the most correct and easiest way to use all the previous work, and found an elegant formulation for this - and then, since he found a very correct tool, he could go far beyond the known results.”

Flying over the jungle



Peter Scholze in June, at a seminar on geometry at Bronn University

Despite the complexity of perfectoid spaces, Scholze is famous for the clarity of his reports and works. “I didn't understand anything until Peter explained to me,” Weinstein says.

According to Karayani, Scholze is trying to explain his ideas at a level accessible even to freshmen. “It gives a sense of openness and generosity of ideas,” she says. “And he does this not only with a bunch of senior mathematicians - a large number of young people have access to it.” According to Karajani, the friendly and open manner of Scholze makes him an ideal leader in his field. Once, when he and Scholze made a difficult trip over rough terrain, “it was he who ran around and made sure that everything was in place and checked everyone,” says Karayani.

But, according to Helman, even after Scholze’s explanations, it’s hard for other researchers to understand the perfectoids. “Get off the path proposed by Scholze, and you will find yourself in the jungle, where everything is very difficult.” But Scholze himself “would never get lost in the jungle, because he does not fight them. He always looks in perspective to see the general concept. ”

Scholze is not entangled in vines, because he forces himself to fly over them: just like in college, when he preferred to work without making notes. He says that this means having to formulate your ideas in the simplest possible way. "The capacity of your head is limited, so you won’t be able to do too complicated things in it."

While other mathematicians are just beginning to deal with perfectoid spaces, some of the most far-reaching discoveries in this area, which is not surprising, were made by Scholze and his coauthors. The result, which was published in 2013, “stunned the community,” according to Weinstein. "We did not even imagine that such a theorem could appear."

Scholze’s result expanded the scope of the rules, known as reciprocity laws, that govern the behavior of polynomials using modulo arithmetic (or hourly arithmetic is not necessarily 12-hour). Modulo arithmetic (in which, for example, 8 + 5 = 1, if the dial has 12 hours) is the most natural and popular finite-number system in mathematics.

The laws of reciprocity - a generalization of the law of reciprocity of quadratic residues, discovered 200 years ago. This is the cornerstone of number theory, and one of Scholze's favorite theorems. The law states that for two primes p and q, in most cases p will be a full square in modular arithmetic modulo q, when q will be a full square in modular arithmetic modulo p. For example, 5 is a full square on a dial with 11 hours (in modular arithmetic modulo 11), since 5 = 16 = 4 2 , and 11 is a full square on a dial with 5 hours, since 11 = 1 = 1 2 .

“This is unexpected for me,” says Scholze. “At first glance, these two things are not related to each other.” According to Weinstein, "most of the modern algebraic number theory can be represented as attempts to generalize this law."

In the mid-twentieth century, mathematicians discovered an incredible connection between the laws of reciprocity and a completely seemingly different field - the hyperbolic geometry of patterns, such as the famous Escher angels / devils tiles.



This connection is the central part of the Langlands Program, a set of interconnected hypotheses and theorems concerning the interrelations of number theory, geometry and analysis. When hypotheses can be proved, they turn out to be very powerful tools: for example, the proof of Fermat’s Great Theorem is based on the solution of one small (albeit non-trivial) part of the Program.

Mathematicians gradually realized that the Langlands program extends far beyond the hyperbolic disc; it can also be studied in higher order hyperbolic spaces and in many other contexts. Scholze showed how to extend it to an extensive set of structures in the "hyperbolic three-space" - a three-dimensional analogue of a hyperbolic disk - and beyond. Having constructed the perfectoid version of the hyperbolic three-space, Scholze discovered a whole set of new reciprocity laws.

“Peter’s work has completely changed the idea of ​​what can be done and what we can achieve,” says Karayani. Weinstein says the Scholze result shows that the Langlands program is “deeper than we thought ... more systematic and ubiquitous.”

Rewind




Discussing mathematics with Scholze is like consulting an oracle, Weinstein says. “If he says:“ Yes, it will work “, then you can be sure of it. If he says no, he must give up immediately; if he says that he does not know (what happens) well, then you are lucky, you have an interesting task. "

Karajani says working with Scholze is not as difficult as it might seem. When she worked with him, she never had a sense of rush. “As if we always did everything right — somehow we proved the most general theorem possible in the best way, creating the right constructions that shed light on things.”

True, once Scholze was in a hurry - trying to finish work at the end of 2013 before the birth of his daughter. According to him, it is good that he was in a hurry then. “Since then, I haven’t really done anything.”

As a father, he began to be more disciplined about his schedule. But he does not need to specifically give up time for research - he simply fills the gaps between other duties. “Mathematics is my passion. I always want to think about her. " However, he is not inclined to romanticize this passion. When asked what it means to be a mathematician, he hesitated. “That sounds too philosophical.”

He loves privacy and feels uncomfortable with growing fame (for example, in March he became the youngest Leibniz Prize winnergiving 2.5 million euros for further research). “Sometimes it's too much,” he says. “I'm trying to make it not affect my daily life.”

Scholze continues to study perfectoid spaces, and also explores other areas, in particular, algebraic topology - she uses algebra to study forms. “Over the past year and a half, Peter has completely mastered this subject,” says Bhatt. “He changed the methods used by experts to reflect on this topic.”

Bhatt says that other mathematicians experience fear and enthusiasm at the same time when Scholze touches their field of activity. “This means that now the topic will begin to develop very quickly. I am delighted that he works in an area that is in contact with mine, and I can really see how the boundaries of knowledge are moving forward. "

Scholze himself considers his work a simple workout. “For the time being, I am in the phase of studying what is already there, and I simply formulate the knowledge in my own way,” he says. “It does not seem to me that I have already started research.”

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