Additive Combinatorics

我的主要研究领域是加法组合学。这个相对较新的数学领域关注的是具有小和集或积集的群的子集的加法或乘法结构。集合的和集是由该集合的元素的成对和组成的。(在产品集中，我们取成对的产品。)加法组合学与数论、调和分析、遍历理论和理论计算机科学有很强的联系。我对加法组合数学中一些问题背后的几何很感兴趣。 通常的情况是，加性组合数学中的一个问题可以转化为关于某一域上的平面上的曲线点的关联问题。一个经典的例子就是所谓的“和积问题”。算术级数有一个很小的和集，等比级数有一个很小的乘积集，但算术级数和等比级数有非常不同的结构。有没有可能同时找到小套装和小套装呢？答案在很大程度上取决于潜在的场或环。有限域的子域有很小的求和集和积集，但在特征为零的域上或当集合离子域“很远”时，这是不可能的。这个普遍的问题被称为“和积现象”。令人惊讶的是，这个问题可以转化为在场上定义的平面中的点和线之间的关联数量的界限。理解这种联系并给出点-曲线关联次数的有效界限是这项研究提案的核心部分。这样的结果不仅对数学的其他领域，而且对计算机科学都有重要的应用。

My primary field of research is additive combinatorics. This relatively new area of mathematics is concerned about the additive or multiplicative structure of subsets of groups with small sumset or product set. The sumset of a set is formed by the pairwise sums of elements of the set. (In a product set we take the pairwise products.) Additive combinatorics has strong connections to number theory, harmonic analysis, ergodic theory, and theoretical computer science. I am interested in the geometry behind some of the problems in additive combinatorics. It is often the case that a problem in additive combinatorics can be translated to a question about curve-point incidences in a plane over some field. A classical example is the so called "sum-product problem". An arithmetic progression has a small sumset and a geometric progression has small product set, but the arithmetic and geometric progressions have very different structure. Is it possible to find a set with small sumset and small product set at the same time? The answer very much depends on the underlying field or ring. A subfield of a finite field has very small sumset and product set but the same is not possible over fields with characteristic zero or when the set is "very far" from a subfield. The general problem is called the "sum-product phenomenon". Surprisingly the problem can be translated to bounding the number of incidences between points and lines in the plane defined over the field. Understanding such connections and giving efficient bounds on the number of point-curve incidences is a central part of this research proposal. Such results would have important applications not only to other fields of mathematics but to computer science as well.