Millennium Challenge. Just about the complicated

Hi, habra people!

Today I would like to touch upon such a topic as the “millennium tasks”, which for dozens and some hundreds of years have been worrying the best minds of our planet.

After proving the hypothesis (now the theorem) Poincaré Gregory Perelman, the main question that interested many was: “ But what did he actually prove, explain on the fingers? "Taking this opportunity, I will try to explain on the fingers the remaining tasks of the millennium, or at least approach it from another, more realistic side.

Equality of classes P and NP

We all remember from school the quadratic equations that are solved through the discriminant. The solution of this problem belongs to the class P ( P olynomial time) - for it there is a fast (hereinafter the word “fast” is meant as performed in polynomial time) a solution algorithm that is memorized.
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There are also NP -tasks ( N on-deterministic P olynomial time) , the solution of which can be quickly checked by a certain algorithm. For example, checking by computer scanning. If we return to solving a quadratic equation, we will see that in this example, the existing solution algorithm is verified as easily and quickly as it is solved. This suggests a logical conclusion that this task belongs both to one class and to the second.

There are many such tasks, but the main question is, is it possible to quickly and easily solve all or not all tasks that can be easily and quickly checked? Now for some problems, no fast solution algorithm has been found, and it is not known whether such a solution exists at all.

On the Internet, I also met this interesting and transparent formulation:

Suppose you, being in a big company, want to make sure that your friend is also there. If you are told that he is sitting in a corner, it will be enough a split second to look at the truth of the information. In the absence of this information, you will be forced to bypass the entire room, treating guests.

In this case, the question is still the same, is there such an algorithm of actions, thanks to which, even without having information about where a person is, find him as quickly as if he knew where he is.

This problem is of great importance for the most diverse areas of knowledge, but it cannot be solved for more than 40 years.

Hodge Hypothesis

In reality, there are many both simple and much more complex geometric objects. Obviously, the more complex the object, the more time consuming it becomes to study it. Now scientists have invented and with might and main an approach, the main idea of ​​which is to use simple “bricks” with already known properties instead of the object being studied, which are glued together and form its likeness, yes, yes, the designer familiar to everyone since childhood. Knowing the properties of "bricks", it becomes possible to approach the properties of the object itself.

Hodge's hypothesis in this case is associated with some properties of both the "bricks" and objects.

Riemann Hypothesis

We all still know from school primes that are divisible only by themselves and by one (2,3,5,7,11 ...) . Since ancient times, people have been trying to find a pattern in their placement, but so far so far no one has smiled at anyone. As a result, scientists applied their efforts to the prime number distribution function, which shows the number of primes less than or equal to a certain number. For example, for 4 - 2 prime numbers, for 10 - already 4 numbers. Riemann's hypothesis just sets the properties of this distribution function.

Many statements about the computational complexity of some integer algorithms are proved under the assumption of the validity of this hypothesis.

Yang-Mills Theory

The equations of quantum physics describe the world of elementary particles. Physicists Yang and Mills, finding a connection between geometry and elementary particle physics, wrote their equations uniting the theories of electromagnetic, weak, and strong interactions. At one time, the Yang-Mills theory was considered only as a mathematical sophistication that had no relation to reality. However, later the theory began to receive experimental confirmation, but in general terms it still remains unsolved.

On the basis of the Yang-Mills theory, a standard model of elementary particle physics was constructed within the framework of which the acclaimed Higgs boson was predicted and recently discovered.

Existence and smoothness of solutions of the Navier – Stokes equations

Fluid flow, air flow, turbulence. These, as well as many other phenomena, are described by equations known as the Navier – Stokes equations . For some particular cases, solutions have already been found in which, as a rule, parts of the equations are discarded as not affecting the final result, but in the general form the solutions of these equations are unknown, and it is not even known how to solve them.

Birch's hypothesis - Swinnerton-Dyer

For the equation x ² + y ² = z ² , Euclidean gave a complete description of solutions at one time, but for more complex equations, the search for solutions becomes extremely difficult; it suffices to recall the history of the proof of Fermat's famous theorem to verify this.

This hypothesis is related to the description of algebraic equations of degree 3 - the so-called elliptic curves and in fact is the only relatively simple general method for calculating the rank, one of the most important properties of elliptic curves.

In the proof of the Fermat theorem, elliptic curves occupied one of the most important places. And in cryptography, they form a whole section of the name of themselves, and some Russian digital signature standards are based on them.

Poincare conjecture

I think if not all, then the majority have heard about it. Most often found, including on the mainstream media, such a transcript as “a rubber band stretched over a sphere can be smoothly pulled into a point, and stretched over a bagel — not ”. In fact, this formulation is valid for the Thurston hypothesis, which summarizes the Poincaré hypothesis, and which Perelman actually proved.

A special case of the Poincaré conjecture tells us that any three-dimensional manifold without an edge (the universe, for example) is like a three-dimensional sphere. And the general case translates this statement to objects of any dimension. It is worth noting that a bagel, just like the universe is like a sphere, is like an ordinary coffee mug.

Conclusion

Currently, mathematics is associated with scientists who have a strange appearance and are talking about equally strange things. Many talk about its isolation from the real world. Many people of both younger and quite conscious age say that mathematics is an unnecessary science, that after school / institute, it was not useful anywhere in life.

But in reality this is not so - mathematics was created as a mechanism with the help of which one can describe our world, and in particular, many observable things. She is everywhere, in every home. As said V.O. Klyuchevsky: "It is not the fault of the flowers that the blind cannot see them."

Our world is far from being as simple as it seems, and mathematics, in accordance with this, is also becoming more complex and improving, providing an increasingly solid ground for a deeper understanding of the existing reality.