Unsolvable problems are 7 interesting mathematical problems. Each of them was proposed at one time by famous scientists, as a rule, in the form of hypotheses. For many decades, mathematicians around the world have been racking their brains over their solution. Those who succeed are awaited by a reward of one million American dollars, offered by the Clay Institute.
Background
In 1900, the great German universal mathematician David Hilbert presented a list of 23 problems.
Research carried out with the aim of solving them 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 unresolved or resolved partially remained:
- the problem of consistency of arithmetic axioms;
- general law of reciprocity in the space of any numerical field;
- mathematical study of physical axioms;
- study of quadratic forms with arbitrary algebraic number coefficients;
- the problem of rigorous justification of the calculus geometry of Fedor Schubert;
- and so forth
Unexplored are: the problem of extending to any algebraic domain of rationality the well-known Kronecker theorem and the Riemann hypothesis .
Clay Institute
Under this name is known private non-profit organization, whose headquarters is in Cambridge, Massachusetts. It was founded in 1998 by Harvard mathematician A. Jeffy and businessman L. Clay. The aim 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 Institute of Mathematics offered a prize to those who solve problems that are known as the most difficult unsolvable problems, naming their list Millennium Prize Problems. From the “Hilbert List” only the Riemann hypothesis entered into it.
Millennium Challenges
The Clay Institute list originally included:
- hypothesis of Hodge cycles;
- the equations of the quantum theory of Yang - Mills;
- Poincare conjecture;
- the problem of 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 realizations.
What proved Grigory Perelman
In 1900, the famous scientist-philosopher Henri Poincare suggested that any simply connected compact 3-dimensional manifold without an edge is homeomorphic to a 3-dimensional sphere. Its evidence in the general case has not been found for a century. Only in 2002-2003 did the St. Petersburg mathematician G. Perelman publish a series of articles with a solution to the Poincare problem. They produced the effect of an exploding bomb. In 2010, the Poincare conjecture was excluded from the list of “Unsolved problems” of the Clay Institute, and Perelman himself was asked 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 was able to prove can be given by imagining that a rubber disk is pulled on a donut (torus), and then they try to pull the edges of its circle to one point. Obviously this is not possible. Another thing, if you make this experiment with a ball. In this case, it would seem that a three-dimensional sphere obtained from a disk whose circumference was pulled to a point by a hypothetical cord will be three-dimensional in the understanding of an ordinary person, but two-dimensional from the point of view of mathematics.
Poincare suggested that the three-dimensional sphere is the only three-dimensional “object”, the surface of which can be pulled together at one point, and Perelman managed to prove this. 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 has the following form: for any simple compact gauge group, a quantum spatial theory created by Yang and Mills exists, and at the same time has a zero mass defect.
If we speak a language that is understandable to an ordinary person, 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 to explain all these interactions. The Yang-Mills theory is a mathematical language, with the help of which it has become possible to describe 3 of the 4 basic forces of nature. It is not applicable to gravity. Therefore, one cannot assume that Young and Mills managed to create a field theory.
In addition, the nonlinearity of the proposed equations makes them extremely difficult to solve. At 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 flow, fluid flow, and turbulence. For some special cases, analytical solutions to the Navier-Stokes equation have already been found, but so far no one has managed to do this for the general. At the same time, numerical modeling for specific values of speed, density, pressure, time, and so on allows you to achieve excellent results. It remains to hope that someone will be able to apply the Navier-Stokes equations in the opposite direction, i.e., calculate the parameters with their help, or prove that there is no solution method.
Birch-Swinnerton Dyer Problem
The category “Unsolved problems” also includes a hypothesis proposed by English scientists from the University of Cambridge. Even 2300 years ago, the ancient Greek scientist Euclid gave a complete description of the solutions of the equation x2 + y2 = z2.
If for each of the primes we count the number of points on the curve modulo it, we get an infinite set of integers. If we specifically “glue” it into 1 function of a complex variable, then we get the Hasse – Weil zeta function for a third-order curve, denoted by the letter L. It contains information about the behavior modulo all primes at once.
Brian Birch and Peter Swinnerton-Dyer put forward a hypothesis regarding elliptic curves. According to her, the structure and quantity of the set of its rational decisions are connected with the behavior of the L-function in unity. The currently unproven hypothesis of Birch-Swinnerton-Dyer depends on the description of algebraic equations of degree 3 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 standards of digital signature are based on their application.
Equality of classes p and np
If the remaining “Millennium Tasks” are purely mathematical, then this is relevant to the current theory of algorithms. The problem regarding the equality of classes p and np, also known as the Cook-Levin problem, can be formulated in clear language as follows. Suppose that a positive answer to a certain question can be checked quickly enough, i.e., in polynomial time (PV). Then is it true that the answer to it can be quickly found? Even simpler, this task sounds like this: is the solution to the problem really no harder to find than finding it? If the equality of the classes p and np is ever proved, then all the selection problems can be solved in the PV. At the moment, many experts doubt the truth of this statement, although they can not prove the opposite.
Riemann hypothesis
Until 1859, no regularity was revealed that would describe how primes are distributed among natural numbers. Perhaps this was due to the fact that science dealt with other issues. However, by the middle of the 19th century, the situation had changed, and they became one of the most relevant, which mathematics began to study.
The Riemann hypothesis that appeared during this period is an assumption that there is a certain regularity in the distribution of primes.
Today, many modern scholars believe that if it is proved, then many fundamental principles of modern cryptography will have to be reviewed, which form the basis of a significant part of e-commerce mechanisms.
According to the Riemann hypothesis, the nature of the 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 primes. For example, there is the problem of “twins,” the difference between which is 2. These numbers are 11 and 13, 29. Other primes form clusters. These are 101, 103, 107 and others. Scientists have long suspected that such clusters exist among very large primes. If they are found, then the strength of modern crypto keys will be in question.
The Hodge Cycle Hypothesis
This unsolved problem is still formulated in 1941. The Hodge hypothesis suggests the possibility of approximating the shape of any object by “gluing” together simple bodies of greater dimension. 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 is hoped that in the near future they will be resolved, and their practical application will help humanity enter a new round of technological development.