Ergodic theory is an area of mathematics which has evolved over the last 100 years to help to describe how transformations and groups act on spaces. This programme will focus on its applications to other areas of mathematics. This is a particular case of the Oppenheim conjecture, which was posed by Sir Alexander Oppenheim in 1929 and proved in the affirmative by G. Margulis.
Ergodic theory is also important in studying the geometry of manifolds with negative curvature.
In a standard approach, one studies the natural action of the fundamental group on the ideal boundary. Studying the ergodic properties of such actions has played a fundamental role in studying negatively curved spaces appearing, for example, in the work of G. Mostow on rigidity, and continues to give new insights into problems in geometry. A large part of the programme will devoted to studying rigidity properties for the natural generalization of flows to higher rank actions.