2006 年以来，美国科学院院士Sarnak陆续提出了两个意义深远的纲领性猜想，旨在将素数分布的所有问题纳入一个统一的框架里，并期望问题有一个统一的解决方案。刘建亚和Sarnak提出了将高维自守形式理论应用于素数分布的一个途径，从而在素数分布领域的Sarnak 猜想的研究中取得实质性进展，作为例子证明了Sarnak猜想在三元二次型的情形对殆素数成立。Sarnak猜想引起了菲尔兹奖得主Bourgain等著名数学家的兴趣，已成为主流数论问题之一。由于这一方向日新月异的迅猛发展，一些重要结果得以证明。本项目计划联合国内外优秀的中青年数学家，针对Sarnak猜想，联合攻关，集中开展定期的研讨活动。

Since 2006, Sarnak (member of the National Academy of Sciences USA) has proposed two far-reaching programmatic conjecture on the distribution of primes. It aims to unify all problems on prime numbers into the same frame, and then try to find a unified solution. Liu and Sarnak developed a new approach of applying the theory of high-dimensional automorphic forms to the distribution of primes, which led to a substantial progress for the study of Sarnak conjecture for the distribution of primes, and proved, as an example, the Sarnak conjecture with almost-prime variables in the context of ternary quadratic forms. Sarnak's conjecture appeals to many famous mathematicians including Bourgain (the Fields Medal winner) , and has become one of the main problems in number theory. Due to the rapid development of this field, some remarkable results have been established.This project plans to organize an advanced seminar on Sarnak's conjecture through the cooperation of outstanding domestic and foreign mathematicians.