August 1900 was marked by the holding of the II International Congress of Mathematicians in Paris, at which one of the leading figures of science, David Hilbert, formulated the most cardinal problems that need to be resolved. Hilbert's 23 problems determined many of the key directions in the development of mathematics in the last century.

By the beginning of the 21st century, almost all of them had been solved, or left the list for other reasons - for example, as vaguely formulated - and a hundred years after Hilbert, mathematician Stephen Smale put forward a new list of 18 problems facing mathematicians and physicists of our time. Smale's attempt can be counted, but the alternative version proposed by the authoritative American Clay Institute has received much more fame. Seven problems were named at a high-profile event specially organized in Paris. One of them, the Riemann hypothesis, migrated from the list of 1900, and another, the Poincaré hypothesis, was proved two years later.

Naked Science presents a brief overview of the Millennium Challenges, each of which the Clay Institute is willing to pay out a million dollars for. By the way, this also applies to Poincaré's hypothesis: the deserved million is still awaiting payment, and so far Grigory Perelman refuses to accept the award. Of course, we simplified many points by trying to explain the problems in such a way that the essence was understandable even to a person who is very far from both higher and any other mathematics.

## 1. Equality of classes P and NP

Field: theory of algorithms

Supposed in the early 1970s, remains unresolved

Imagine that you need to purchase office equipment, furniture and stationery for 500 thousand rubles - and you are looking at the supplier's price list. You can choose what you want, but the list must include two printers, one executive chair, 50 ballpoint pens, the rest is optional. How many combinations are possible? This is a variant of the "knapsack problem", which in the classical form consists in putting as many things of a certain volume and value as possible in the volume of a backpack. It is easy to check the final version, but it is difficult to find it. These tasks, by the way, include "cracking" someone else's password, which is encrypted in such a way that the system can easily check its correctness, but it is almost impossible for a cracker to figure out the correct version in the sea of alternative solutions to an encrypted string.

Such problems in the theory of algorithms belong to the NP complexity class: their solution can be quickly verified. Some of them belong to the class P - those whose solution is also easy to find (in "foreseeable", or, more strictly speaking, polynomial time). The question is whether there always exist simple algorithms for solving NP-problems - that is, whether the classes NP and P are equal. Today it is assumed that the answer to it will be negative: not all problems whose solutions are easily testable can be easily solved. NASA mathematician Subit Chakrabarti predicts that a definitive answer may be obtained within the next 50 years.

## 2. Existence and smoothness of solutions of the Navier - Stokes equations

Field: mathematical physics (hydrodynamics)

The problem has been known for over a hundred years, remains unsolved

The problem at the junction of mathematics and classical physics grows out of work done back in the 19th century, when scientists began to formulate strict laws that describe the movement of fluids.The Navier - Stokes equations obtained at that time remain one of the most important in hydrodynamics and aerodynamics. They allow you to calculate the flow rate taking into account viscosity, compressibility, density, pressure, etc., and are used everywhere. However, it is still not possible to solve them in general form, and calculations are carried out only for individual, special cases.

The solution of the Navier-Stokes equations hides many secrets of one of the "hardest nuts" of modern physics - the problem of turbulence. With it, modern technology is ubiquitous, from airplanes and submarines to wind farms and cars - but much of the turbulence remains poorly understood, poorly calculated and almost unpredictable. Therefore, scientists are storming this Millennium Challenge with particular persistence. Mathematician Subit Chakrabarti suggests that within half a century, a solution to complex turbulence equations can be found.

In the meantime, Kazakhstani mathematician Mukhtarbai Otelbayev submitted an application for victory, in whose calculations an error was subsequently found, as well as Uzbek scientist Shokir Dovlatov, whose solution is still being verified. American mathematician Stephen Smale is the recipient of the Fields Prize for his work in topology. In 2000, he was head of the mathematics department at the University of California at Berkeley, when Academician Vladimir Arnold, then president of the International Mathematical Union (IMU), suggested that he pick up a list of new problems to replace the already completed Hilbert list. Smale picked 18 such problems, some of which, including the equality of P and NP, the Poincaré and Riemann hypotheses, solutions of the Navier-Stokes equations, etc., were also included in the list of Millennium Problems prepared by the Clay Institute.

## 3. Riemann hypothesis

Field: number theory

Formulated in 1859, remains unresolved

Many of us from school remember the existence of prime numbers - those that are divisible only by 1 and by themselves, like 2, 3, 5, 7, 11, etc. Prime numbers play an important role in the "abstract" theory numbers, and in practice - for example, in the work of cryptographic algorithms. If we mark the position of all primes on the numerical axis, then we will see that their distribution is uneven and, it seems, does not obey some pattern, therefore, it is impossible to predict in advance exactly where the next prime number will appear. However, Bernard Riemann showed that this distribution is similar to the points at which the zeta function - ς (s) = 1 / 1s + 1 / 2s + 1 / 3s + 1 / 4s +… - vanishes.

It is known that it has zero value when s is a negative even number. But where else? According to Riemann's calculations, other zeros appear if s is a complex number containing the real part 1/2. The problem was named among the topical ones by David Hilbert in 1900 and has not yet been solved, although almost all mathematicians are ready to agree: calculations carried out even with the use of supercomputers and for incredibly huge primes confirm the validity of the Riemann hypothesis. It has been proven for about 10 trillion first decisions, but not in general terms yet. According to Subit Chakrabarti, over the years of work on this problem, mathematicians have advanced quite far, and the answer can be found in the coming decades.

## 4. Poincaré's hypothesis

Scope: topology

Appeared in 1900-1904, resolved in 2002.

Poincaré's hypothesis refers to topology, one of the most difficult and young areas of mathematics, which studies the properties of geometric figures and their deformations that occur without discontinuity. Mold a pyramid from plasticine - you can easily turn it into a cone, cylinder or even a sphere, without gluing or tearing anything anywhere. Blind a bagel - and you will not succeed in such a trick, although the bagel is easily deformed, for example, into a cup with a handle. Strictly speaking, the surfaces of the sphere and the cylinder are homeomorphic, while the spheres and the torus are non-homeomorphic.But this is for the simplest case: Poincaré showed that any closed (without holes) two-dimensional surface is homeomorphic to a two-dimensional sphere. The solution for surfaces of higher dimensions took about a century.

Interestingly, for dimensions 5 and higher, Poincaré's conjecture was proved back in the 1960s, and for dimension 4 - in the 1980s. The case where any three-dimensional surface is homeomorphic to a three-dimensional sphere turned out to be the most difficult. This was shown only in 2002 by the St. Petersburg mathematician Grigory Perelman, who instantly became famous all over the world. After a series of unpleasant intrigues and attempts to take away the glory of the discoverer from him, Perelman, who already had the fame of a "crazy genius", broke all contacts with the official mathematical world, refused to receive a cash prize from the Clay Institute and leads a reclusive lifestyle, not accepting numerous proposals for work and participation in all kinds of professional events and forums.

## 5. Hodge hypothesis

Field: algebraic geometry

Formulated in 1941, remains unresolved

Since the time of Descartes, algebraic geometry has made great progress in describing the shapes of complex objects. We can propose an equation whose solutions will correspond to a particular figure, for example, describe a sphere as (x - a)^{2} + (y - b)^{2} = r^{2}… If the object is too complex, we can approximate this shape by "gluing" together simpler shapes - then it will correspond to the solution of a system of equations. This approach is used very widely, and mathematicians have gone far from even objects that generally correspond to any geometric analogs - to what is called the broader term "manifold".

The question is to what extent this approach can be applied to a special class of projective algebraic varieties. The Scotsman William Hodge found an ingenious method for checking the correspondence of such varieties and the algebraic equations of their representation, but it has not yet been possible to prove its validity in the general case. Moreover, mathematician Subit Chakrabarti considers this problem too "abstract" for the current level of development of science - its solution requires the development of new, poorly mastered sections of algebraic geometry, and will be found very soon. So far, the hypothesis has been proven only for some special cases, and mathematicians do not know whether it is true in principle.

## 6. Young - Mills theory

Field: mathematical physics (physics of elementary particles)

Originated in 1950s, remains unresolved

The Yang - Mills theory belongs to the field of elementary particle physics, being the foundation of modern ideas about them. Basically, it is a set of equations that try to predict particle behavior and is an attempt to provide a unified description of three of the four fundamental interactions of nature - strong, weak, and electromagnetic. This was only partially achieved by creating an apparatus for describing the combined electroweak interaction. It is not yet possible to solve the equations by including strong interaction in them, and a separate solution has been found for it, which, by the way, led to the discovery of quarks.

It turns out that the Yang - Mills theory includes the electroweak interaction and - separately - the strong one. Experiments show that it, in principle, can unite them: the predictions of the equations agree with experiments, both natural and calculated, model. However, this has not yet been mathematically proven. It is shown that such a rigorous theory requires constructing descriptions for each compact gauge group - that is, a group of transformations in which the properties of the particle system remain unchanged (as the phase shift does not affect the properties of the electron wave), and this must be done for a four-dimensional space. time. Subit Chakrabarti suggests that solving this problem will take about a century and the elaborate work of several generations of mathematicians.

## 7.Birch - Swinnerton-Dyer hypothesis

Field: algebraic geometry

The problem was put forward in the early 1960s, remains unsolved

Equations in which both variables and solutions are integers are named Diophantine after the ancient Greek mathematician. In their simplest form, they are really simple - like, for example, x2 = y: we remember that the geometric solution of such a school equation will be a parabola. But in more complex cases, things get really complicated. Moreover, even the Soviet mathematician Yuri Matiyasevich showed that there is no universal solution to Diophantine equations, thereby answering the question of Hilbert's 10th problem.

The Birch - Swinnerton-Dyer hypothesis (these are two people - Peter Swinnerton-Dyer and Brian Birch) asserts that the set of solutions to the elliptic curve is related to the behavior of the L-function in region 1. This function is calculated as the zeta function already familiar to us from the Riemann hypothesis, and the number of rational solutions is infinite if (and only if) when L (1) = 0. Mathematician Viktor Kolyvagin proved in one direction that if L (1) ≠ 0, then the number of rational points is finite. There is no way to do the reverse calculations. According to Subit Chakrabarti, it is possible that the final proof of this hypothesis, in principle, cannot be obtained, as evidenced by the answer to the question of Hilbert's 10th problem. Probably, answers to the Birch - Swinnerton-Dyer conjecture will be obtained only in a private form.