In his 28 years, Peter Scholz reveals the deep connections between the theory of numbers and geometry

In 2010, a startling rumor spread through the community of people who study number theory, and it came to Jared Weinstein . Allegedly some graduate student from the University of Bonn in Germany published a paper in which 288-page proof of the theorem from number theory is only up to 37 pages. 22-year-old student Peter Scholz found a way to get around one of the most difficult parts of the proof by comparing the theory of numbers and geometry.

“It’s just incredible 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.”
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Mathematicians from the University of Bonn, who assigned Scholz the title of professor just two years later, already knew about his extraordinary mental abilities. After the publication of the work, experts began to notice it both in the theory of numbers and in geometry.

Since then, Scholze, now 28, has grown to a high position 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 say about him as a favorite among the applicants for the Fields Prize, one of the highest awards for the mathematician.

The key innovation of Scholz - the class of fractal structures, which he called the perfectionoid spaces - has just turned a few years, but it already leads to far-reaching consequences in the field of arithmetic geometry, in which number theory and geometry merge. Weinstein says that Scholze’s work was prophetic. "He could see the consequences before they began to happen."

Bargav Bhatt [Bhargav Bhatt], a mathematician at the University of Michigan who wrote collaborations with Scholze, says that many mathematicians react to his work "with a mixture of awe, fear and arousal."

And this is not because of his character, which colleagues describe as down to earth and generous. “He never lets you know that he is superior to you,” says Eugene Hellmann, a university colleague from Scholz. Rather, it is because of its frightening ability to look so deep into the essence of a mathematical problem. Unlike many mathematicians, he begins his work not with a specific task that requires solving, but with some elusive concept, which he wants to understand for the sake of interest. But then, according to Anna Caraiani , an expert on number theory from Princeton University, who worked with Scholz, the constructions he created “find applications in a million other directions that were not originally predictable - simply because the correct objects.

Learning arithmetic

Mathematical Institute at the University of Bonn, Germany

Scholze began to independently comprehend the institute mathematics at the age of 14, attending the Heinrich Hertz gymnasium, 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, Scholz learned that ten years earlier, Andrew Wiles had proved the famous 17th century theorem, known as the Great Fermat Theorem , which states that the equation x ⁿ + y ⁿ = z ^{n has} no solutions in integers greater than zero for n> 2. Scholze really wanted to study the proof, but it quickly turned out that, despite the simplicity of the theorem, its proof uses the mathematics of the most advanced level. “I didn't understand anything, but it was very cool,” he says.

And Scholze began to study what gaps in knowledge he needed to fill in order to understand this evidence. “And usually I still teach everything,” he says. “I never studied basic things like linear algebra — I learned them by studying something else.”

Buzzing into evidence, he was struck by mathematical objects called modular forms and elliptic curves that mysteriously uncover such disparate fields 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.

The mathematical tastes of Scholze began to be determined. Today, he still has problems with simple equations and integers. And these tangible roots quite clearly make him feel even the esoteric mathematical structures. “In fact, I am fond of arithmetic,” he says. According to him, he is happier when his abstract designs lead him back to small discoveries associated with ordinary integers.

After graduation, Scholze continued to study number theory and geometry at the University of Bonn. As his classmate Helman recalls, Scholze did not write anything down in his math class. Hellman claims that Scholze understood the course material in real time. "I did not 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 such 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 voids between integers and fractions. But this system is based on a non-standard idea about the location of these voids and the proximity of numbers to each other. In a p-adic system, two numbers are close not when the difference between them is small, but when the difference between them is divided by the degree of p (the higher the degree, the closer the numbers).

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

The p-adic numbers “are far away from household intuition,” says Scholze. But over the years they have become natural to him. “Now for me, real numbers are more complicated than p-adic ones. I got used to them so much that the material seems to me much more strange. ”

In the 1970s, mathematicians noticed that many problems on p-adic numbers become easier if these numbers are expanded by an infinite tower of number systems, in which each one is wrapped around the bottom p times, and p-adic numbers are at the bottom of this tower. "Upstairs" of the infinite tower is a wrapping space - a fractal object, which is the simplest example of perfect perfection spaces that Scholz would later develop.

Scholze set himself the task of understanding why these infinite wrapping constructions simplify so much 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 is possible to create perfect space for a variety of mathematical structures. He showed that these spaces make it possible to move the questions related to polynomials from the world of p-adic numbers to other mathematical areas where arithmetic is greatly simplified (for example, do not make transfer when adding). “The strangest property of perfect spaces is that they can magically move between two numerical systems,” says Weinstein.

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

Hellman 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."

Flight over the jungle

Peter Scholz in June at a seminar on geometry at the University of Bronnian

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

According to Karayani, Scholze tries to explain his ideas at a level that is accessible even to first-year students. “It gives a sense of openness and generosity of ideas,” she says. “And he is doing this not only with a handful of senior mathematicians - a large number of young people have access to him.” As Karayani says, Scholze’s friendly and open style makes him the ideal leader in his field. Once, when they, together with Scholze, made a difficult march across 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 is difficult for other researchers to understand the perfect-play spaces. “Depart from the trail proposed by Scholze, and you will find yourself in the jungle, where everything is very difficult.” But Scholze himself “would never be lost in the jungle, because he does not fight with them. He always looks in perspective to see the general concept. ”

Scholze does not get tangled up in vines, because he forces himself to fly over them: much like in college, when he preferred to work without making notes. He says that this means having to formulate your ideas in the simplest way. "The capacity of your head is limited, so too complicated things in it will not work."

While other mathematicians are just beginning to deal with perfect spaces, one of the most far-reaching discoveries in this area, which is not surprising, was made by Scholz and his co-authors. The result, which was published in 2013, "led the community into a stupor," as Weinstein says. “We didn’t even imagine that such a theorem could appear.”

Result Scholz expanded the scope of the rules, known as reciprocity laws, governing the behavior of polynomials using modular arithmetic (or hour arithmetic is not necessarily 12-hour). Arithmetic modulo (in which, for example, 8 + 5 = 1, if the dial has 12 hours) is the most natural and popular system for studying finite numbers in mathematics.

The laws of reciprocity - a generalization of the law of reciprocity of quadratic residues, discovered 200 years ago. It 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 full square in modular arithmetic mod q, when q is full square in modular arithmetic mod p. For example, 5 is a full square on the dial with 11 o'clock (in modular arithmetic modulo 11), because 5 = 16 = 4 ² , and 11 is a full square on the dial with 5 o'clock, because 11 = 1 = 1 ² .

“For me, this is unexpected,” 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 seemingly completely different area - the hyperbolic geometry of patterns, such as the famous tiles of Escher's angels / devils.



This relationship is the central part of the Langlands Program, a set of related hypotheses and theorems concerning the interrelationships of number theory, geometry, and analysis. In the case when the 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 extended far beyond the hyperbolic disk; 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 built a perfect version of a hyperbolic three-space, Scholze discovered a whole new set of reciprocity laws.

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

Rewind

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

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

True, one day Scholze was in a hurry, trying to finish the job 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 done much."

Becoming 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 voids 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. "It sounds too philosophical."

He loves privacy, and feels uncomfortable with increasing fame (for example, in March he became the youngest winner of the Leibniz Prize , giving 2.5 million euros for further research). “Sometimes it's too much,” he says. “I’m trying to make sure that it doesn’t 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 of thinking about this topic that are used by experts.”

Bhatt says that other mathematicians feel fear and excitement at the same time when Scholze deals with their field of activity. “This means that now the topic will start to develop very quickly. I am delighted that he is working in an area in contact with mine, and I really see how the boundaries of knowledge are moving forward. ”

Scholze himself considers his work a simple warm-up. “I am still in the phase of studying what is already there, and I just formulate knowledge in my own way,” he says. “It does not seem to me that I have already begun to engage in research.”