The proposed research is in the area of smooth dynamics and ergodic theory. The main focus of the proposed research is the study of dynamical systems with multidimensional time. Intensive research during the past two decades proved that wide variety of such systems which display certain degree of chaotic behavior, are remarkably rigid. These results induced fast progress towards some long standing conjectures in number theory and quantum mechanics. Rigidity of such systems stands in sharp contrast with flexibility of chaotic systems with one-dimensional time. There are two main themes within the proposed research. The first is to explore further stability and rigidity of algebraic multidimensional-time systems with less chaotic behavior: existence and local rigidity (i.e. differentiable stability) of partially hyperbolic abelian actions on nilmanifolds, and local rigidity of parabolic abelian actions on certain classes of locally symmetric spaces. In this direction the proposed research involves the KAM theory approach and thus requires a detailed study of the corresponding infinitesimal problem: the description of the first cohomology over these actions. The second theme is to explore the existence and stability properties of non-algebraic systems which are strongly partially hyperbolic. Such systems have specific structure of invariant foliations which on one hand imposes restrictions for the manifold of the action, and on the other hand it tends to be robust under small perturbations thus leading to certain degree of stability for the action. The research in this direction leads towards global classification of strongly partially hyperbolic multidimensional-time systems.
Dynamical systems and chaos are the areas of mathematics which have flourished during past years while maintaining a strong connection with their roots which lie in the study of various phenomena in domains like cell biology, nano technology, meteorology and engineering. The studies of evolution of nature systems in time represent one of the core scientific interests today. The problem of stability of systems is one of the main issues which arises in the study of nature systems as mathematical models are merely approximations of the natural phenomena. In celestial mechanics, stability of the solar system is one important topic, and the KAM theory turned out to be a powerful tool towards better understanding of the system's long term behavior. Systems with multi-dimensional time appear in quantum mechanics, where rigidity of such systems is present in the form of uniform distribution of quantum states. Multidimensional-(lattice) time systems also appear in the mathematical formalism for quasicrystals, whose physical properties and generation have been intensively studied. In hardware architecture one of the recently explored venues is extending the classical methodology to multidimensional time (multidimensional scheduling). The principal investigator will continue to encourage undergraduate students, female in particular, to take active part in the research process and will make an effort to expose them to various aspects and applications of this research. These activities will benefit from the grant.
