Dynamical Aspects of Ramsey Theory

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The mathematical analysis of dynamical systems -- systems which evolve through time -- has found diverse applications in fields from economics and weather forecast to engineering and astronomy. Many systems arising from such applications tend to exhibit chaotic behavior and can be very difficult to analyze directly. Nevertheless, it is always possible to obtain useful information about such systems by studying their long term behavior. The overarching goal of this project is to develop new methods to study the long term behavior of dynamical systems in a theoretical setting and to explore applications to the mathematical subject of combinatorics.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 phenomena of multiple recurrence and convergence of multiple ergodic averages along polynomials and other natural classes of sequences.