What connects the theory of numbers with the path of light?

Original author: Kevin Hartnett
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Following Gauss, we recognize the “royal” status of mathematics, and given that our company has a competence center “Algorithmic Support”, we often have interesting materials on this topic: our colleagues write their own, author’s articles, then, studying that interesting things happen to foreign colleagues, prepare brief reviews and translations of third-party articles. It will probably be useful to those who share our interests, so we decided to share these materials and knowledge.

In mathematics it often happens that it is incredibly difficult to understand the simplest things that seem to be known to everyone, such as rational numbers. For example, the search for rational solutions of Diophantine equations of mathematics has been around for hundreds of years. The ideas borrowed from physics helped to get closer to solving a thousand-year problem. We present an article published in Quanta Magazine, with our partial translation and overview.

Minhyun Kim, a mathematician at the University of Oxford, is trying to figure out which rational numbers allow certain types of Diophantine equations to be solved. This mathematical problem, according to general estimates, is about 3000 years old. Since rational decisions do not obey geometric patterns, this is indeed a difficult task. So complicated that Gerd Faltings received the Fields award in 1986 only for proving that some classes of Diophantine equations have a finite number of rational solutions. The mathematicians themselves call the Faltings breakthrough “ineffective evidence” because it does not call the exact number of rational decisions and does not allow them to be identified.

Kim tries to consider rational numbers in an expanded numeric context in which hidden patterns begin to appear. Kim was able to find such a context in physics: according to mathematics, rational solutions have a lot in common with the trajectory of light. For a long time Kim doubted that he was right and that his work could convince other scientists and only recently did he decide to present his idea to the general public. According to Kim himself, over the next 15 years, number theory will become much more closely intertwined with physics.

Kevin Hartnett , author of an article published in Quanta, writes:

“Mathematicians often say that the more symmetrical an object is, the easier it is to study it. With this in mind, they would like to place the study of Diophantine equations in a more symmetrical context than the one in which the problem usually appears. If this can be done, it would be possible to use the detected symmetry to find the necessary rational points.

Sets of numbers can also be symmetric, and the more symmetric the set of numbers is, the easier it is to understand: you can use symmetric relations to calculate unknown values. Numbers with a certain type of symmetric relationship form a “group”, and you can use the properties of a group to understand all the numbers in it. But the set of rational solutions of the equation does not have symmetry and does not form a group, which leaves mathematicians alone with the impossible task, trying to find all solutions one by one.

From the 1940s, mathematicians began to explore ways of placing Diophantine equations in more symmetrical contexts. Claude Chabati discovered that within a larger geometric space that he built using p-adic numbers, rational numbers form their own symmetric subspace. He combined this subspace with the graph of Diophantine equations: their intersection points correspond to rational solutions of the equation.

In the 1980s, Robert Coleman completed the work of Chabati. For several decades after this, the Coleman-Chabati approach was the best tool for finding rational solutions to Diophantine equations. However, it only works when the equation graph is in a certain proportion with respect to a larger space. When this proportion does not meet the requirements, the search for exact points at which the equation curve intersects with rational numbers becomes more complicated.

Kim, in order to expand the work of Chabati, wanted to find an even larger space into which Diophantine equations could be placed. ”

And here Kim proposes to use an analogue of the physical concepts of "space-time", "space of spaces":

“To understand why, consider a ray of light. Physicists believe that light moves through the multidimensional field space. In this space, the light will move along a path that corresponds to the principle of “least action”, that is, along a path that minimizes the time required to move from point A to point B. This principle explains why light is refracted when moving from one medium to another: curved path minimizes elapsed time. Such larger spaces of spaces encountered in physics possess additional symmetries that are absent in all the spaces that they represent. These symmetries draw attention to certain points, emphasizing, for example, the time-minimizing path. Constructed in a different way or in a different context, these same symmetries may emphasize other points, for example, points,

In number theory, there is something like spacetime. This something also offers various ways of forming paths and constructing the space of all possible paths. Kim develops a scheme in which the problems of finding the trajectory of light and finding rational solutions to Diophantine equations are the facets of one problem.

Solutions of Diophantine equations form spaces, curves, which are given by equations. These curves can be one-dimensional, like a circle, or multi-dimensional. For example, if you plot the complex solutions of the Diophantine equation x4 + y4 = 1, you get a torus with three holes. The rational points of this torus do not have a geometric structure, this makes their search a complex task, but they can correspond to points in a more multidimensional space of spaces that will already have a certain structure. ”

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