Why we hold this seminar?

Understanding of Yonghui Wang

300 BC, Euclid may have been the first to give a serious proof on the distribution of primes, which asserts that there are infinitely many primes.  In 21st century, The newest great result on primes is due to Ben Green and Terence Tao,  whose work has been regarded as the most important breakthrough in this area after Chen Jinrun's work on Goldbach Type problems (1973, over 30 years later).  As a reward, the second author won the Fields Medal in 2006 with this work in the first position of his honoured list.

Green-Tao's theorem generalizes the Euclid's theorem on primes,  states that, for any given positive integer k, there are infinitely many prime points, Here, points means the k-vectors. And prime points means that the coordinates of  the k-vectors are  all primes. In their theorem, the k-vectors must take the explicit form of

(n,n+r,...,n+(k-1)r)  for any n,r positive integer.

That is, the coordinates consists of an arithmetic progression of length k.  But, it is natural to expect that more theorems on the distribution of primes can be found in the other appropriate forms.

There are several genuine ingredients in their works. Indeed, they give an Ergodic view of prime number theorems, which contains the classical prime number theorem (Chebyshev's) as a special case. As we know, the classical prime number theorem is the starting point of any study of primes, e.g. Goldbach conjecture. So, we can expect that Green-Tao's theorem will be a new starting point for the further study of primes.

The schema of Green-Tao's theorem, Lectured by Wang Yonghui