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But that was not clear. They must analyze special functions, called Type I and Type II sums, for each type of their problem, and then show that the sums were the same regardless of which constraints they used. Only then will Green and Sawhney know that they can replace their evidence without losing knowledge.
They soon realized that: They could show that the money was the same by using a tool that each of them had come across in a previous job. This device, known as Gowers, was invented many years ago by a mathematician Timothy Gowers measure how a function or number is random or ordered. On the face of it, Gowers’ culture seemed to be that of an entirely different mathematician. Sawhney said: “It is impossible as an outsider to say that these things are related.”
But using the results confirmed in 2018 by mathematicians Terence Tao and Tamar ZieglerGreen and Sawhney found a correlation between Gowers’ behavior and Type I and II statistics. Instead, they needed to use Gowers’ values to show that their two sets of foundations – those built using solid sculptures, and those built using real foundations – were sufficiently similar.
As it turned out, Sawhney knew how to do this. Earlier this year, in order to solve the problem of inconsistency, he developed a method to compare sets using Gowers’ method. Surprisingly, the method was good enough to show that the two groups had type I and type II statistics.
With this in hand, Green and Sawhney confirmed what Friedlander and Iwaniec thought: There are many primitives that can be written as. p2 + 4q2. In the end, they were able to expand their results to confirm that there are many primes for other types as well. The results are very successful for a problem that is often very rare.
More importantly, this work shows that Gowers’ habit can be a powerful tool in new environments. “Because it’s new, maybe in this part of number theory, there’s a lot of potential to do more,” Friedlander said. Mathematicians are now hoping to expand Gowers’ culture further – trying to use it to solve other problems in number theory beyond basic arithmetic.
“It’s really exciting for me to see things that I thought about before have unexpected new applications,” Ziegler said. It’s like a parent, when you release your child and they grow up and do amazing and unexpected things.
Original article reprinted with permission from Quanta Magazineindependent publication of Simons Foundation whose mission is to advance the public’s understanding of science by explaining what is happening in mathematical and physical science research.
