Pongalai's Conjecture is one of the seven mathematical conundrums of the century, along with Riemann's postulate, Hodge's Conjecture, and Yang-Mills theory.

One of the Seven Problems of the Century: P (polynomial algorithm) versus NP (non-polynomial algorithm) problem

Two of the Seven Problems of the Century: Hodge's conjecture

Three of the Seven Problems of the Century: Pongalai's Conjecture

Pongalai's Conjecture

The Seven Problems of the Century -Yang-Mills Existence and Mass Gap

Conundrum #6: Existence and Smoothness of Navier-Stokes Equations

The "Thousand Happinesses Problem" No. 1: P (Polynomial Algorithm) Problem vs. NP (Non-Polynomial Algorithm) Problem

On a Saturday night, you attend a big party. Feeling rushed, you wonder if there is anyone in this hall that you already know. Your host suggests to you that you must know the lady, Rose, who is in the corner near the dessert tray. It does not take a second for you to scan there and realize that your host is correct. Without such a suggestion, however, you would have to look around the entire hall and examine each person, one by one, to see if there is anyone you recognize. Generating a solution to a problem usually takes much more time than verifying a given solution. This is an example of this general phenomenon. In a similar vein, if someone tells you that the numbers 13, 717, and 421 can be written as the product of two smaller numbers, you may not know whether to believe him, but if he tells you that it can be factored out as 3607 times 3803, then you can easily verify that this is correct with a pocket calculator. Regardless of the dexterity with which we write programs, determining whether an answer can be verified very quickly using inside knowledge, or whether there is no such hint and it takes a lot of time to solve, is seen as one of the most salient problems in logic and computer science. It was stated by Stephen Cook in 1971.

The Hodge conjecture

Twentieth-century mathematicians found a powerful way to study the shape of complex objects. The basic idea was to ask to what extent we could form the shape of a given object by gluing together simple geometrically constructed blocks of increasing dimension. This technique was to become so useful that it could be generalized in many different ways; eventually leading to some powerful tools that allowed mathematicians to make great progress in classifying the diverse objects they encountered in their research. Unfortunately, in this generalization the geometrical starting point of the program becomes blurred. There is a sense in which certain components must be added that have no geometric interpretation whatsoever. The Hodge conjecture asserts that for a particularly perfect type of space known as projective algebraic clusters, the components called Hodge closed chains are in fact (rational linear) combinations of geometric components called algebraic closed chains.

Three of the "Thousand Cries Puzzle": The Poincare Conjecture

If we stretch a rubber band around the surface of an apple, we can make it move slowly and contract to a point without either tearing it or letting it leave the surface. On the other hand, if we imagine that the same rubber band is stretched over a tire tread in the proper direction, then there is no way to shrink it to a point without tearing the band or the tire tread. We say that the surface of an apple is "monoconnected" while the surface of a tire is not. About a hundred years ago, Poincaré already knew that a two-dimensional spherical surface can be essentially characterized by monoconnectivity, and he posed the problem of the correspondence of three-dimensional spherical surfaces (the totality of points in four-dimensional space that are at a unit distance from the origin). The problem immediately became immensely difficult, and mathematicians have been struggling with it ever since.

The Riemann Hypothesis

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers. Such numbers are called prime numbers; they play an important role in both pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern; however, the German mathematician Riemann (1826~1866) observed that the frequency of prime numbers is closely related to the disposition of a carefully constructed so-called Riemann Cetta function z(s$. The famous Riemann hypothesis asserts that all meaningful solutions of the equation z(s)=0 lie on a straight line. This has been verified for the beginning 1,500,000,000 solutions. Proving that it holds for every meaningful solution would bring light to many of the mysteries surrounding the distribution of prime numbers.

The Millennium Puzzle #5: Yang-Mills Existence and the Mass Gap

The laws of quantum physics hold for the world of elementary particles in the same way that the Newtonian laws of classical mechanics hold for the macroscopic world. About half a century ago, Yang and Mills discovered that quantum physics reveals a striking relationship between the physics of elementary particles and the mathematics of geometric objects. The predictions based on the Yang-Mills equations have been confirmed in high-energy experiments fulfilled in laboratories around the world as follows: at Brockhaven, Stanford, the European Institute of Particle Physics, and Tsukuba. Nevertheless, their equations, which both describe heavy particles and are mathematically rigorous, have no known solutions. In particular, the "mass gap" hypothesis, recognized by most physicists and applied in their explanations of the invisibility of "quarks", has never been mathematically satisfactorily confirmed. Progress on this issue requires the introduction of fundamentally new ideas, both physical and mathematical.

The Sixth Problem of the Thousand-Year Puzzle: The Existence and Smoothness of the Navier-Stokes Equations

Rising and falling waves follow our boats meandering in lakes, and turbulence follows the flight of our modern jets. jet airplanes in flight. Mathematicians and physicists are convinced that both breezes and turbulence can be explained and predicted by understanding the solutions to the Navier-Stokes equations. Although these equations were written in the 19th century, our understanding of them remains minimal. The challenge is to make substantial advances in mathematical theory that will allow us to unlock the mysteries hidden in the Navier-Stokes equations.

Seventh of the "Thousand and One Problems": The Birch and Swinnerton-Dyer conjectures

Mathematicians have always been fascinated by the problem of carving out all the integer solutions to algebraic equations such as x^2+y^2=z^2. problems. Euclid used to give complete solutions to this equation, but for more complex equations this becomes extremely difficult. In fact, as Yu.V. Matiyasevich pointed out, the Hilbert tenth problem is unsolvable, i.e., there exists no general method for determining whether such a method has an integer solution. When the solution is a point of an abelian cluster, the conjecture of Behe and Svinaton-Dyer holds that the size of the group of rational points is related to the disposition of an associated Caietta function z(s) in the neighborhood of the point s = 1. In particular, this interesting conjecture holds that if z(1) is equal to 0,then there exists an infinite number of rational points (solutions), and conversely, if z(1) is not equal to 0,then there exists only a finite number of such points.