Dynamical aspects of Ramsey theory

Overview Inspired by Furstenberg's proof of Szemeredi's theorem using ergodic theory in 1977, several results in Ramsey theory and combinatorial number theory were since obtained using ideas and methods from dynamical systems. The interplay with combinatorics has likewise enriched the fields of ergodic theory and topological dynamics with a wealth of interesting new problems. The two main research directions of this project are: 1. To develop new techniques within the framework of dynamical systems to solve Ramsey theoretic problems, with emphasis on those concerning partition regularity of (systems of) non-linear equations. 2. To improve the understanding of the long term behaviour of measure preserving systems by exploring the phenomenons of multiple recurrence and convergence of multiple ergodic averages along polynomials and other natural classes of sequences. Intellectual Merit This project seeks to address, and potentially settle, some interesting conjectures and problems in Ergodic Ramsey Theory. This area lies at the crossroads of several mathematical disciplines, such as combinatorics, number theory, Fourier analysis, ergodic theory, topological dynamics and topological algebra; and any progress will likely reveal new aspects of the rich and fascinating connections between them. Broader Impacts The mathematical analysis of dynamical systems has found diverse applications in fields from economics and weather forecast to engineering and astronomy. Most of the systems which arise from such applications exhibit chaotic behaviour and therefore are too difficult to analyse locally. In such cases, one can still obtain useful information by studying the long term behaviour of the system at hand. This project will improve the toolbox available to perform such analysis and, in this way, will ultimately impact a number of other fields.