A large unified mathematical theory just got a little closer
“We are more likely to believe that all guests are right, but seeing it is really exciting,” said Anna Karaiani, a mathematician at the Imperial College of London. “And in the case you really thought it was out of reach.”
This is just the beginning of the hunting that takes years – the mourning eventually wants to show the modular lentils for each level. But the result can now help answer many open questions, just as the modular proof of elliptical curves has opened up all kinds of research.
Through the glass to follow
The elliptical curve is a particular type of equation that uses only two variables –x Vat LetterIf you chart the solutions, you will see that the curves look simple. But these solutions are intertwined in rich and complex ways, and in many of the most important questions the theory of number appears. For example, the speculation of Birch and Swinnerton-Dyer-one of the most difficult problems in mathematics, with a $ 1 million reward for anyone who proves it-is about the nature of elliptical curves.
The elliptical curves can be studied directly. So sometimes mathematicians prefer to approach them from another angle.
This is where the modular shapes enter. A modular form is a highly symmetrical function that appears in an apparently separate area of mathematical study called analysis. Because they show excellent symmetry, modular forms can work with them easier.
Initially, these objects seem to be not related. But the proof of Taylor and Wales showed that every elliptical curve corresponds to a particular modular form. They have specific properties – for example, a set of numbers that describe elliptical curved solutions is also modular. Therefore, mathematicians can use modular forms to gain new insight into elliptical curves.
But mathematicians think the modular theorem of Taylor and Wales is just an example of a universal reality. There is a much more general class than objects beyond elliptical curves. And all of these objects must also have a wider world of symmetrical functions such as modular forms of partner. This, in essence, is what the Langlands program is about.
The elliptical curve only has two variables –x Vat Letter– So it can be drawn on a flat paper. But if you add another variable, ZYou receive a curved surface that lives in 3D space. This more sophisticated object is referred to as a Abelian surface, and like elliptical curves, its solutions have an ornamental structure that mathematicians want to understand.
Natural it seemed that the water levels should match the more sophisticated types of modular shapes. But the additional variable makes them much harder to build and it is much harder to find solutions. They also appeared to meet a modular theorem, it seemed completely out of reach. “It was a well -known problem that you didn’t think about, because people were thinking about it and stuck,” Jay said.
But boxers, calories, Jay and Qarha’i wanted to try.
Finding a bridge
“All four mathematicians were involved in research into the Langlands program, and they wanted to prove one of these speculations for” an object that actually appears in real life, and not weird. “
Not only does the Ablian levels appear in real life – the real life of a mathematician, that is, proving a modular theorem for them opens new mathematical doors. “If you have the sentence that you have no chance to do illegal, you can do a lot of things,” Calry said.
The mathematicians collaborated in 2016, in the hope that Taylor and Wales had the same steps in proven their elliptical curves. But each of these steps was much more complex for the Abelian levels.
So they focused on a particular type of Abelian surface, called a conventional ablican surface, which was easier to work with. For each level, there is a set of numbers that describes the structure of its solutions. If they can indicate that the same set of numbers can also be obtained from a modular form, they are done. These numbers act as a unique label and allow them to pair each of their Ablian surfaces with a modular form.
