The Green-Tao theorem for number fields
- March 22 at 16:00 - 18:10, 2021 (JST)
- Dr. Wataru Kai (Assistant professor, Mathematical Institute, Tohoku University)
- via Zoom
5, 11, 17, 23, 29 are prime numbers which form an arithmetic progression of length 5. A famous theorem of Ben Green and Terence Tao in 2008 says there are arbitrarily long arithmetic progressions of prime numbers. Algebraic number theorists are also interested in more general numbers like square roots of integers. Recently, Mimura, Munemasa, Seki, Yoshino and I have established a generalization of the Green-Tao theorem in such a direction.
In the first 50 minutes of my talk, I would like to explain some background and technology behind the Green-Tao theorem. In the second half after a break, I explain the concept of number fields to formulate our generalization of their result. I will also discuss how one of the new difficulties, which I call the norm vs length conflict, is handled by a technique called Geometry of Numbers.
*Please contact Keita Mikami or Hiroyasu Miyazaki's mailing address to get access to the Zoom meeting room.