Some problems in additive combinatorics

申请人计划研究加性组合学中的各种问题，具体来说：继续他在Polymath4项目上的工作，以产生一个快速找到大素数的确定性算法；探索开创性的概率方法的分支，他开发了一种方法来解决某些加法组合问题（如二维角问题）；继续发展一些关于和积不等式的新思想。一个很好的例子是提议者在Polymath4项目中的工作：到目前为止，他和其他参与者已经产生了一个算法来计算质数计数函数的奇偶性，比以前的任何方法都快；此外，提议者已经产生了该算法的“多项式环模拟”，它显示了完成该项目的巨大希望（当进一步推广时，也许）。理解质数（2、3、5、7、11等）的结构是古希腊人不朽的遗产之一，近年来取得了巨大进展。Polymath4是一个由Timothy Gowers、Gil Kalai和Terrence Tao发起的在线合作研究项目，它解决了一个相关的、基本的问题：一个人产生大素数的速度有多快？提议者一直是这个项目的主要参与者，并且迄今为止在解决这个问题方面取得了重大进展。他计划继续与Polymath4的其他研究人员和他的一些学生合作。此外，他计划继续他的工作，在一个相对较新的领域开发一些突破性的方法，称为“加性组合学”，由于高尔斯、格林、陶和其他人的工作，这个领域最近已经成为一个热门领域。

The proposer plans to work on a variety of problems in additive combinatorics,specifically: Continuing his work on the Polymath4 project to produce a deterministic algorithm to find large prime numbers quickly; exploring the ramifications of a ground-breaking probabilistic approach that he has developed to attack certain additive combinatorial problems (such as the 2D corners problem); and continuing the development of some fresh ideas on sum-product inequalities. A good example here is the proposer's work in the Polymath4 project: To date he, and other participants, have produced an algorithm to compute the parity of the prime counting function faster than any previous method; and furthermore, the proposer has produced a ``polynomial ring analogue'' of this algorithm which shows great promise towards completing the project (when generalized further, perhaps).Understanding the structure of prime numbers (2,3,5,7,11, and so) is one of the enduring legacies of the ancient Greeks, and in recent years enormous progress has been made. Polymath4, an online collaborative research program initiated by Timothy Gowers, Gil Kalai, and Terrence Tao, addresses a related, fundamental question: How quickly can one even produce large prime numbers? The proposer has been a major participant in this project, and has thus far made significant strides in addressing this problem. He plans to continue working on it with other Polymath4 researchers and with some of his students. In addition, he plans to continue his work on developing some ground-breaking methods in a relatively new field called ``additive combinatorics'', which has become a hot area of late due to works of Gowers, Green, Tao, and others.