A great uniform theory of mathematics approached a little

“We often believe that all guesses are correct, but it is extremely exciting to see that they are already fulfilled,” he said, he said KarabianMathematics scientist at the Imperial College in London. “And if I really thought they would be out of reach.”

It is just the beginning of her hunting that will take years – they eventually want presidential scientists to show the unit of every ABELIAN surface. But the result can actually help answer many open questions, just as the unit has been proven to the elliptical curves opened all kinds of new search trends.

Through the look of the appearance

The elliptical curve is a basic type of equation that uses only two variables –x and Y. If you draw their solutions, you will see what it seems to be simple curves. But these solutions are interconnected in rich and complex ways, and they appear in many of the most important numbers theory questions. For example-one of the most difficult open problems in mathematics, with a bonus of one million dollars for those who prove this first-about the nature of solutions for elliptical curves.

East curves can be difficult to study directly. So sometimes mathematicians prefer to deal with them from a different angle.

This is where normative forms come. The standard form is a very symmetrical function that appears in an apparent separate area of sports study called analysis. Since it shows many pleasant symmetries, normative models can be easier to work with them.

Initially, these things look as if they should not be linked. But Taylor and Wales’s guide revealed that each elliptical curve corresponds to the shape of a specific unit. They have certain common properties – for example, a group of numbers that describe solutions for the elliptical curve that will also appear in their associated standard form. Therefore, mathematicians can use normative forms to gain new visions in the curves of Al -Elilji.

But mathematicians believe that the theory of Taylor and Wales is just one case of a global reality. There is a more general category of organisms that exceed elliptical curves. All of these things must have a partner in a wider world of similar functions such as normative forms. This, in essence, is what is the Langlands program.

The elliptical curve contains only two variables –x and Y– So that it can be drawn on a flat sheet of paper. But if you add another variable, ZYou get a zigzag surface that lives in a three -dimensional area. This most sophisticated object is called Abelian, and as with elliptical curves, its solutions have a decorative structure that mathematicians want to understand.

It seemed natural that the Apilla surfaces are compatible with more complex types of normative forms. But the additional variable makes them more difficult to create and solutions more difficult to find it. Proof that, also, they satisfy the theory of units that seemed completely out of reach. “It was a problem known not to think about it, because people thought about it and commented,” said Ji.

But Boxer, Calegari, Gee and Pilloni wanted to try.

Find a bridge

“All the four mathematicians participated in research on the Langlands program, and they wanted to prove one of these guesses of” an object that already appears in real life, rather than something strange, “Caligari said.

Abilian surfaces do not appear in real life – the true life of the mathematician, that is – but the typical theory of it will be proven will open new mathematical doors. “There are a lot of things that you can do if you have this statement that you have no chance to do otherwise,” said Caligari.

Mathematics began working together in 2016, hoping to follow the same steps that Taylor and Wales had in their evidence of elliptical curves. But each of these steps was more complicated for the surfaces of Abelian.

So they focused on a specific type of Abelian, which is called the regular Abelian surface, was easy to work with. For any such surface, there is a set of numbers that describe the structure of their solutions. If they can show that the same set of numbers can also be derived from a standard form, it will be done. The numbers will serve as a unique sign, allowing them to associate each of their ABELAN surfaces with a standard shape.