New trends in mathematical fluid dynamics and ergodic number theory

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Many phenomena in nature are connected with deep mathematical problems and theories. As an example, one can mention the obvious fact that in many regions on the Earth the frequency of tornadoes has strongly increased. Tornadoes are special solutions of equations of fluid dynamics that have strong singularities. A big part of the mathematical efforts detailed in this proposal is connected with the general properties of solutions of equations of fluid dynamics. Several years ago D. Li and I proposed a new approach which allows us to construct solutions which have singularities that in some respects resemble tornadoes. Recently a group of researchers from Italy performed numerical studies and demonstrated solutions that fully confirmed our theory. Another part of the proposal is connected with number theory. One of the main objects in number theory is the so-called Moebius function. P.Sarnak has proposed a wide program on Mobuius functions whose aim was the analysis of statistical properties of the Moebius functions. Some progress was achieved in our joint papers with F.Cellarosi and with M. Avdeeva and Dong Li. A new feature here is the appearance of probability distributions with complex probabilities which certainly is a new phenomenon. The purpose of the proposal is to study in more detail complex probabilities, new limit theorems and find other applications to number theory.The topics in this proposal center around questions in the fields of ergodic theory and the theory of partial differential equations. In broad terms, ergodic theory is the study of the asymptotic behavior of the paths of objects in motion. This field was borne through the study of statistical mechanics, and indeed ergodic theory remains a valuable tool in this important subject of physics. Another subject in which ergodic theory has found application is in number theory, and in particular in the study of the asymptotic distribution of certain kinds of numbers. The application of ergodic theory to number theory has itself led to fascinating questions in ergodic theory. This proposal contains a collection of such questions that we will continue to study. Partial differential equations are the principal tool by which physical phenomena, such as the flow of air over the wing of an airplane, the behavior of plasmas in the body of a star, the flow of electrons through a semi-conductor, are modeled. It is a subject which is intensively studied for its own inherent interest as well as for its usefulness in applications. We propose the continuation of deep studies in the properties of the equations of fluid dynamics as we described above.