The sum-product phenomenon in various groups, expanding maps and applications

不同群体中的和积现象，扩展地图和应用

• 批准号: 0700297
• 负责人: Mei-Chu Chang
• 金额: $ 17.24万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700297&HistoricalAwards=false
• 关键词: sum; product; phenomenon; various; groups

The proposal belongs to a research activity in combinatorial number theory, in particular 'additive combinatorics'. Many of the problems go back quite a while, to the work of Erdos-Szemeredi and Freiman. But the revitalization of part of this subject appears mainly during the last decade, with motivations from harmonic analysis such as the Kakeya set problem in higher dimension, the work on arithmetic progressions of Gowers and Green-Tao, the 'sum-product theorem' in finite fields and its many applications (to computer science, the theory of exponential sums and to the spectral theory of Heckeoperators) . The problems in the proposal are mainly related to the sum-product and product phenomena in various rings and groups, continuing a line of research (in particular by the PI) that turned out to have rather unexpected applications besides their intrinsic interest from the combinatorial point of view. Indeed, purely combinatorial (and basically elementary) techniques brought progress on issues that had stalled for quite some time, such as on the expander properties of certain 'thin' SL2 Caley graphs, and the 'expanding properties' of algebraic functions on finite fields.Recent results around the 'sum-product' and 'product-phenomena' in various fields, rings and groups lead to progress on an amazing number of issues, ranging from computer science to representation theory. The purpose of the proposal is to explore further several types of questions that underly these applications, including those are of importance to quantum computation and nanotechnology.

该提案属于组合数论的研究活动，特别是“加法组合学”。许多问题可以追溯到相当长的一段时间，埃尔多斯-塞梅雷迪和弗赖曼的工作。但振兴的一部分，这一主题主要出现在过去十年中，与动机从谐波分析，如挂谷集问题在更高的层面上，工作算术级数的高尔斯和绿色道，“总和产品定理”在有限领域及其许多应用（计算机科学，理论的指数和频谱理论的Heckeoperators）。该提案中的问题主要与各种环和群中的和积和积现象有关，继续了一系列研究（特别是PI的研究），这些研究结果除了从组合的角度来看其固有的兴趣之外，还有相当意想不到的应用。事实上，纯粹的组合（基本上是基本的）技术在已经停滞了相当长一段时间的问题上取得了进展，例如某些“薄”SL 2 Caley图的扩展属性，以及有限域上代数函数的“扩展属性”。最近在各个领域，环和群中围绕"和-积“和”积-现象“的结果导致了惊人数量问题的进展，从计算机科学到表征理论。该提案的目的是进一步探讨这些应用背后的几种类型的问题，包括那些对量子计算和纳米技术具有重要意义的问题。