Aspects of Harmonic Analysis and Hamiltonian PDEs

Institute For Advanced Study, Princeton NJ

Investigators

Abstract

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.