Unsolvable problems: the Navier-Stokes equations, the Hodge conjecture, the Riemann hypothesis. The Millennium Goals

Unsolvable tasks are 7 most interesting mathematical problems. Each of them was proposed in due time by famous scientists, usually in the form of hypotheses. For decades, over their decision, they have been breaking the heads of mathematics all over the world. Those who succeed will be rewarded with the million-dollar award offered by the Clay Institute.

Prehistory

In 1900, the great German mathematician-general David Gilbert, presented a list of 23 problems.

Studies carried out to solve them have had a huge impact on the science of the 20th century. At the moment, most of them have already ceased to be riddles. Among the unsolved or resolved partially remained:

• The consistency of arithmetical axioms;
• The general law of reciprocity on the space of any number field;
• Mathematical study of physical axioms;
• The study of quadratic forms for arbitrary algebraic number coefficients;
• The problem of rigorous justification of the calculus geometry of Fedor Schubert;
• And others.

The following are unexpected: the problem of extending to any algebraic domain the rationality of the well-known Kronecker theorem and the Riemann hypothesis .

Clay Institute

Under this name is known a private non-profit organization, whose headquarters are in Cambridge, Massachusetts. It was founded in 1998 by Harvard mathematician A. Jeffey and businessman L. Clay. The purpose of the Institute is to popularize and develop mathematical knowledge. To achieve this, the organization issues awards to scientists and sponsors promising research.

At the beginning of the 21st century, the Clay Mathematical Institute offered an award to those who solve problems that are known as the most difficult unsolvable problems, calling their list the Millennium Prize Problems. From the "List of Hilbert" only the hypothesis of Riemann entered it.

The Millennium Goals

The list of Clay Institute originally included:

• Hypothesis of Hodge cycles;
• The equations of the quantum Yang-Mills theory;
• The Poincare conjecture ;
• The problem of equality of classes P and NP;
• The Riemann hypothesis;
• Navier Stokes equations, on the existence and smoothness of its solutions;
• The Birch-Swinnerton-Dyer problem.

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

What has proved Grigory Perelman

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

The most understandable explanation of what the Russian mathematician managed to prove is given by imagining that a rubber bag is being pulled onto a donut (tor), and then they try to pull the edges of its circle into one point. Obviously, this is impossible. Another thing, if you make this experiment with the ball. In this case, it would seem that the three-dimensional sphere obtained from the disk, whose circumference is pulled into the point by a hypothetical cord, will be three-dimensional in the understanding of the ordinary man, but two-dimensional in terms of mathematics.

Poincare suggested that the three-dimensional sphere is the only three-dimensional "object", the surface of which can be pulled into one point, and Perelman managed to prove this. Thus, the list of "Unsolvable tasks" today consists of 6 problems.

The 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 space theory created by Yang and Mills exists and 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 be a tool to explain all these interactions. The Yang-Mills theory is a mathematical language, with the help of which it became possible to describe 3 of the 4 basic forces of nature. It does not apply to gravity. Therefore, it can not be assumed that Yangu and Mills succeeded in creating a field theory.

In addition, the nonlinearity 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 clear how these equations can be solved for a strong coupling.

The Navier-Stokes equations

With the help of these expressions, processes such as air currents, flow of liquids and turbulence are described. For some particular cases, analytic solutions of the Navier-Stokes equation have already been found, but no one has yet succeeded in doing this for the general. At the same time, numerical simulation for specific values of speed, density, pressure, time, and so on allows you to achieve excellent results. It is to be hoped that someone will be able to apply the Navier-Stokes equations in the opposite direction, that is, calculate the parameters using them, or prove that there is no solution method.

The Birch-Swinnerton-Dyer problem

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

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

Brian Birch and Peter Swinnerton-Dyer put forward a hypothesis concerning elliptic curves. According to it, the structure and the number of its rational solutions are related to the behavior of the L-function in the unit. The Birch-Swinnerton-Dyer conjecture, which has not yet been shown, depends on the description of algebraic equations of the third degree and is the only relatively simple general method for calculating the rank of elliptic curves.

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

The equality of the classes p and np

If the remaining "Millennium Challenges" are purely mathematical, then this relates to the current theory of algorithms. The problem concerning the equality of the classes p and np, also known as the Cook-Levin problem, can be formulated in a clear language in the following way. Suppose that a positive answer to a certain question can be checked quite quickly, that is, in a polynomial time (PV). Then is the statement correct that the answer to it can be found quite quickly? Even simpler, this problem sounds like this: is it really possible to check the problem solution more easily than to find it? If the equality of the classes p and np is ever proved, then all the selection problems can be solved for PV. At the moment, many experts doubt the truth of this statement, although they can not prove the opposite.

The Riemann hypothesis

Until 1859, there was not revealed any pattern that would describe how simple numbers are distributed among natural numbers. Perhaps this was due to the fact that science was engaged in other issues. However, by the middle of the 19th century the situation had changed, and they had become one of the most relevant, which mathematics began to deal with.

The Riemann hypothesis that appeared in this period is the assumption that there is a definite regularity in the distribution of primes.

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

According to Riemann's conjecture, the nature of the distribution of prime numbers may be significantly different from what is supposed to be at the moment. The fact is that so far no system has been discovered in the distribution of prime numbers. For example, there is a problem of "twins", the difference between which is equal to 2. These numbers are 11 and 13, 29. Other primes form clusters. This is 101, 103, 107, etc. Scientists have long suspected that such clusters exist among very large prime numbers. If they are found, the resilience of modern crypto-keys will be in question.

Hypothesis about cycles of Hodge

This unresolved problem has been formulated in 1941. Hodge hypothesis suggests the possibility of approximating the shape of any object by "gluing together" simple bodies of greater dimension. This method was 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 in the near future they will be resolved, and their practical application will help mankind to enter a new stage of technological development.