Aspects of Harmonic Analysis and Hamiltonian PDEs

PI打算继续他的研究在组合数论，指数和频谱问题周围的扩展和赫克运营商。其中一部分是与A. Gamburd和P. Sarnak。一个相关的问题是Furstenberg的不变测度问题和各种群作用的刚度矩阵，PI计划与E。林登施特劳斯PI还参与了与去随机化问题和伪随机序列理论相关的加法组合学和指数和的其他应用。该提案是五年前开始的一系列研究的进一步发展，发现了围绕所谓的"和-积现象"的某些新的和非常基本的组合原理。 近年来，它的重要性变得越来越明显，因为它被证明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的进展。围绕扩展器的研究成为真正的跨学科和兴趣的研究人员在各个领域也以外的数学。例如，扩展器的新的鲁棒构造在经由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作用。

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.