Harmonic analysis and arithmetic combinatorics

调和分析与算术组合

• 批准号: 229818-2007
• 负责人: Laba, Izabella
• 金额: $ 2.55万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=478760
• 关键词: Harmonic; analysis; arithmetic; combinatorics

Arithmetic combinatorics is an emerging area of mathematics which combines problems, ideas and methods from harmonic analysis, analytic and combinatorial number theory, combinatorics, and ergodic theory. It is organized around a number of central questions, such as the existence of patterns (e.g. arithmetic progressions) in large sets, or determining the structure of ``extremal" configurations (e.g. sets with small sumsets). Its recent highlights include the work of Gowers on Szemer\'edi's theorem and the Green-Tao result on long arithmetic progressions in the primes. My proposed research concerns the interconnections between harmonic analysis and various aspects of arithmetic combinatorics, in particular harmonic-analytic methods in additive number theory, connections between harmonic analysis and incidence geometry, and arithmetic-combinatorial methods in analysis and geometric measure theory.

算术组合数学是一个新兴的数学领域，它结合了调和分析，分析和组合数论，组合数学和遍历理论的问题，思想和方法。 它是围绕一些中心问题组织的，例如大集合中模式的存在（例如算术级数），或确定“极值”配置的结构（例如具有小和集的集合）。 其最近的亮点包括工作的高尔斯Szemer\'edi的定理和绿陶结果长期算术进展的素数。我建议的研究涉及谐波分析和算术组合学的各个方面之间的相互联系，特别是谐波分析方法在加法数论，谐波分析和关联几何之间的连接，以及算术组合方法在分析和几何测度理论。