Ergodic and Statistical Phenomena in Dynamics

Princeton University, Princeton NJ

Investigators

Abstract

The project consists of two parts. In the first part, the principal investigator will work on various questions related to the appearance of singularities in solutions of the equations of fluid dynamics and other nonlinear partial differential equations. In mathematics this is connected with one of the famous Clay problems. The PI has already established some results in the area. These were obtained with the help of the so-called renormalization group method, which is one of the main methods in the theory of dynamical systems and allows one to study various kinds of singularities of solutions. The project will consider new types of boundary problems and other types of nonlinear partial differential equations. This part of the project might also require some numerical studies. The second part of the project will focus on several problems from ergodic theory related to number theory. This includes the statistics of values of linear forms on small scales and problems related to the famous Littlewood conjecture and the distribution of large Frobenius numbers. One must stress that the ensembles and probability distributions that arise in this context are quite different from the ones in traditional probability theory and require new ideas and new methods. The first major component of the project deals with partial differential equations, which represent one of the principal mechanisms for providing mathematical models of physical phenomena. Of special interest in this project are equations that arise in fluid dynamics and that have application to the study of such traumatic events as hurricanes and tornadoes. The second key component of this project focuses on ergodic theory. Ergodic theory is an area of mathematics in which the subjects of probability and dynamical systems come together. It, too, has applications to real-world phenomena, predictions of weather, climate change, and earthquakes, to mention just a few. Ergodic theory usually develops in conjunction with many important parts of mathematics. It is expected that the progress in ergodic theory that will be made in this project will have an impact on our understanding of many important events in nature and find possible application in other scientific disciplines such as physics and geology.