New Directions in Homogeneous Dynamics and Diophantine Approximation

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This project deals with certain algebraic dynamical systems and their applications to number theory. A dynamical system here stands for an abstract set of points together with an evolution law that governs the way points move over time. Such abstract dynamical systems form the basis for models of a wide range of important phenomena in science and engineering. It turns out that also many problems in mathematics concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of dynamical systems. Furthermore, systems that arise in this context are of algebraic nature, which makes it possible to use a wide variety of sophisticated tools for their investigation. This research project aims to advance the framework of algebraic dynamical systems in approximation theory, develop new methods, and obtain far-reaching generalizations of results in the field. Graduate and undergraduate students will be involved in the project and introduced to new methods and techniques in number theory and dynamics. These students will also supervise research projects within the framework of the PRIMES program, an after-school research program for high school students. During recent years there has been an influx of new ideas concerning connections between number theory and dynamical systems, making it possible to rethink a spectrum of possibilities for further work. This research project continues the study of phenomena in both homogeneous dynamics and Diophantine approximation related to asymptotic properties of orbits using those new ideas and methods. Dynamical tools to be used are: quantitative non-divergence on the space of lattices, Schmidt games and their modifications, recurrence of random walks and integral inequalities, measure rigidity, effective mixing, and equidistribution properties.