Ergodic and Statistical Phenomena in Dynamics

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Three projects are proposed. The first one is on a new approach to the analysis of the dynamics described by the three-dimensional Navier-Stokes system. Such system is being considered as an infinite dimensional dynamical system in Fourier space. The second one is on ergodic properties of internal exchange transformations, in particular the properties of weak mixing of flows on two-dimensional surfaces and their applications, especially to number theory. The last project is on the statistical version of the (3x+1)-problem. The development of mathematical theory of Navier-Stokes system can lead to better understanding of mechanisms of development of singularities in other important non-linear systems of equations, in particular, the prediction of tornados. The (3x+1)-problem will have connections and applications to number theory. The PI will also actively involve in training and mentoring graduate students and postdoctoral associates.