Unsolvable problems are 7 most interesting mathematical problems. Each of them was proposed at one time by well-known scientists, as a rule, in the form of hypotheses. For many decades, mathematicians all over the world have been racking their brains over their solution. Those who succeed will be rewarded with a million US dollars offered by the Clay Institute.

## Backstory

In 1900, the great German mathematician David Hilbert presented a list of 23 problems.

Research carried out to solve them had a huge impact on the science of the 20th century. At the moment, most of them have ceased to be mysteries. Among the unresolved or partially resolved were:

- problem of consistency of arithmetic axioms;
- general law of reciprocity on the space of any number field;
- mathematical study of physical axioms;
- study of quadratic forms for arbitrary algebraic numericalodds;
- the problem of rigorous justification of the computational geometry of Fyodor Schubert;
- etc.

Unexplored are: the problem of extending the well-known Kronecker theorem to any algebraic region of rationality and the Riemann hypothesis.

## The Clay Institute

This is the name of a private non-profit organization headquartered in Cambridge, Massachusetts. It was founded in 1998 by Harvard mathematician A. Jeffey and businessman L. Clay. The aim of the Institute is to popularize and develop mathematical knowledge. To achieve this, the organization gives awards to scientists and sponsors promising research.

At the start of the 21st century, the Clay Mathematics Institute offered a prize to those who solve what are known as the hardest unsolvable problems, calling their list the Millennium Prize Problems. Only the Riemann hypothesis was included in the Hilbert List.

## Millennium Challenges

The Clay Institute's list originally included:

- Hodge cycle hypothesis;
- quantum Yang-Mills theory equations;
- Poincaré hypothesis;
- the problem of the equality of classes P and NP;
- Riemann hypothesis;
- Navier-Stokes equations, on the existence and smoothness of its solutions;
- Birch-Swinnerton-Dyer problem.

These open mathematical problems are of great interest, as they can have many practical implementations.

## What did Grigory Perelman prove

In 1900, the famous philosopher Henri Poincaré suggested that any simply connected compact 3-manifold without boundary is homeomorphic to a 3-dimensional sphere. Its proof in the general case was not found for a century. Only in 2002-2003, the St. Petersburg mathematician G. Perelman published a number of articles with a solution to the Poincaré problem. They had the effect of an exploding bomb. In 2010, the Poincare hypothesis was excluded from the list of "Unsolved Problems" of the Clay Institute, and Perelman himself was offered to receive a considerable reward due to him, which the latter refused without explaining the reasons for his decision.

The most understandable explanation of what the Russian mathematician managed to prove can be given by imagining that a rubber disk is pulled onto a donut (torus), and then they try to pull the edges of its circle into one point. Obviously this is not possible. Another thing, if you make this experiment with a ball. In this case, a seemingly three-dimensional sphere, resulting from a disk whose circumference was pulled to a point by a hypothetical cord, would be three-dimensional in the understanding of an ordinary person, but two-dimensional in terms of mathematics.

Poincare suggested that a three-dimensional sphere is the only three-dimensional "object" whose surface can be contracted to one point, and Perelman managed to prove it. Thus, the list of "Unsolvable problems" today consists of 6 problems.

## Yang-Mills theory

This mathematical problem was proposed by its authors in 1954. The scientific formulation of the theory is as follows:for any simple compact gauge group, the quantum spatial theory created by Yang and Mills exists, and at the same time has a zero mass defect.

Speaking in a language understandable to an ordinary person, the interactions between natural objects (particles, bodies, waves, etc.) are divided into 4 types: electromagnetic, gravitational, weak and strong. For many years, physicists have been trying to create a general field theory. It should become a tool for explaining all these interactions. Yang-Mills theory is a mathematical language with which it became possible to describe 3 of the 4 main forces of nature. It does not apply to gravity. Therefore, it cannot be considered that Yang and Mills succeeded in creating a field theory.

Besides, the non-linearity of the proposed equations makes them extremely difficult to solve. For small coupling constants, they can be approximately solved in the form of a series of perturbation theory. However, it is not yet clear how these equations can be solved with strong coupling.

## Navier-Stokes equations

These expressions describe processes such as air currents, fluid flow, and turbulence. For some special cases, analytical solutions of the Navier-Stokes equation have already been found, but so far no one has succeeded in doing this for the general one. At the same time, numerical simulations for specific values of speed, density, pressure, time, and so on can achieve excellent results. It remains to be hoped that someone will be able to apply the Navier-Stokes equations in reversedirection, i.e. calculate the parameters using them, or prove that there is no solution method.

## Birch-Swinnerton-Dyer problem

The category of "Unsolved Problems" also includes the hypothesis proposed by British scientists from the University of Cambridge. Even 2300 years ago, the ancient Greek scientist Euclid gave a complete description of the solutions to the equation x2 + y2=z2.

If for each prime number we count the number of points on the curve modulo it, we get an infinite set of integers. If you specifically “glue” it into 1 function of a complex variable, then you get the Hasse-Weil zeta function for a third-order curve, denoted by the letter L. It contains information about the behavior modulo all prime numbers at once.

Brian Birch and Peter Swinnerton-Dyer conjectured about elliptic curves. According to it, the structure and number of the set of its rational solutions are related to the behavior of the L-function at the identity. The currently unproven Birch-Swinnerton-Dyer conjecture depends on the description of 3rd degree algebraic equations and is the only relatively simple general way to calculate the rank of elliptic curves.

To understand the practical importance of this task, it is enough to say that in modern cryptography a whole class of asymmetric systems is based on elliptic curves, and domestic digital signature standards are based on their application.

## Equality of classes p and np

If the rest of the Millennium Challenges are purely mathematical, then this one hasrelation to the actual theory of algorithms. The problem concerning the equality of the classes p and np, also known as the Cooke-Levin problem, can be formulated in understandable language as follows. Suppose that a positive answer to a certain question can be checked quickly enough, i.e., in polynomial time (PT). Then is the statement correct that the answer to it can be found fairly quickly? Even simpler this problem sounds like this: is it really not more difficult to check the solution of the problem than to find it? If the equality of the classes p and np is ever proved, then all selection problems can be solved for PV. At the moment, many experts doubt the truth of this statement, although they cannot prove the opposite.

## Riemann Hypothesis

Until 1859, no pattern was found that would describe how prime numbers are distributed among natural numbers. Perhaps this was due to the fact that science de alt with other issues. However, by the middle of the 19th century, the situation had changed, and they became one of the most relevant that mathematics began to deal with.

The Riemann Hypothesis, which appeared during this period, is the assumption that there is a certain pattern in the distribution of prime numbers.

Today, many modern scientists believe that if it is proven, then it will be necessary to revise many of the fundamental principles of modern cryptography, which form the basis of a significant part of the mechanisms of electronic commerce.

According to the Riemann hypothesis, the characterthe distribution of primes may be significantly different from what is currently assumed. The fact is that so far no system has been discovered in the distribution of prime numbers. For example, there is the problem of "twins", the difference between which is 2. These numbers are 11 and 13, 29. Other prime numbers form clusters. These are 101, 103, 107, etc. Scientists have long suspected that such clusters exist among very large prime numbers. If they are found, then the strength of modern crypto keys will be in question.

## Hodge cycle hypothesis

This still unsolved problem was formulated in 1941. Hodge's hypothesis suggests the possibility of approximating the shape of any object by "gluing" together simple bodies of higher dimensions. This method has been known and successfully used for a long time. However, it is not known to what extent simplification can be made.

Now you know what unsolvable problems exist at the moment. They are the subject of research by thousands of scientists around the world. It remains to be hoped that they will be resolved in the near future, and their practical application will help humanity enter a new round of technological development.