Dynamics on homogeneous spaces and Moduli spaces

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Dynamical systems is the study of the evolution of systems which are changing over time. As a concrete example one can consider the trajectory of a ball on an ideal table. The table is frictionless and the angle of incidence equals the angle of reflection. A classical mathematical problem is to study the trajectories of a ball when the sides of the table form a polygon - not necessarily a rectangle. There can be different types of trajectories. Some trajectories can be periodic and some can be dense on the table. This simple problem is notoriously difficult. It is very difficult to solve the problem for a particular table, unless it is of a special shape; for example, a rectangle or an equilateral triangle. This leads to the study of families of tables that have similarities. You could for instance, study the family of tables with five sides. Taking the family of tables as a new space it is possible to define a new flow on this space. This point of view turns out to have important consequences. For example, we may now be able to say what happens on "most" tables. This proposal studies dynamical systems by taking a similar point of view.We mainly seek rigidity results where rather weak initial data about an object yields an almost complete classification of the object. The following will be the main objectives: (i) Employing a dynamical approach to study problems in number theory and geometry has proven rather fruitful. However, this approach is often noneffective. We will seek effectivization of the rigidity phenomena for the action of groups generated by unipotent subgroups on homogeneous spaces; these rigidity results have served as one of the main tools in the aforementioned applications. (ii) There is an action of the group of nonsingular, real, two by two matrices on the moduli space of a compact Riemann surface; this is closely related to the asymptotic of the number of periodic trajectories on rational polygonal tables. This proposal seeks generalizations of the recent exciting developments which proved certain rigidity results for this action. (iii) We attempt to investigate dynamics on homogeneous spaces with infinite volume, and on homogeneous spaces arising from local fields of positive characteristic. There are various geometric and number theoretical applications which motivate the study of these spaces.