Aspects of Harmonic Analysis and Hamiltonian PDEs

调和分析和哈密顿偏微分方程的各个方面

• 批准号: 0808042
• 负责人: Jean Bourgain
• 金额: $ 39.22万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-15 至 2013-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808042&HistoricalAwards=false
• 关键词: Aspects; Harmonic; Analysis; Hamiltonian; PDEs

The PI intends to continue his research in combinatorial number theory, exponential sums and spectral problems around expanders and Hecke operators. Part of this is in an ongoing collaboration with A. Gamburd and P. Sarnak. A related issue is Furstenberg's invariant measure problems and stiffness conjectures for various group actions, which the PI plans to pursue jointly with E. Lindenstrauss. The PI is also involved in other applications of developments around additive combinatorics and exponential sums related to derandomization issues and the theory of pseudo-random sequences.The proposal is a further development of a line of research that started five years ago with the discovery of certain new and quite elementary combinatorial principles around so-called 'sum-product phenomena'. Over the recent years its importance became increasingly clear as it turned out to have a surprisingly broad range of applications (from number theory to computer science) and lead to progress on various problems that were stalled for a long time. Research around expanders became really interdisciplinary and of interest to researchers in various fields also outside mathematics. For instance new robust construction of expanders play a role in connection with building expander-based computer architectures via the process of nanoscale self-assembly.

PI 打算继续他在组合数论、指数和以及围绕扩展器和 Hecke 算子的谱问题方面的研究。其中一部分是与 A. Gamburd 和 P. Sarnak 的持续合作。一个相关的问题是 Furstenberg 的不变测度问题和各种群体行为的刚度猜想，PI 计划与 E. Lindenstrauss 共同研究这些问题。 PI 还参与了与去随机化问题和伪随机序列理论相关的加性组合学和指数和的发展的其他应用。该提案是五年前开始的一系列研究的进一步发展，该研究围绕所谓的“和积现象”发现了某些新的且相当基本的组合原理。 近年来，它的重要性变得越来越明显，因为它具有令人惊讶的广泛应用（从数论到计算机科学），并导致长期停滞的各种问题取得进展。围绕扩展器的研究真正成为跨学科的，并且引起了数学以外各个领域的研究人员的兴趣。例如，新型稳健的扩展器结构在通过纳米级自组装过程构建基于扩展器的计算机架构方面发挥着重要作用。