Dispersive partial differential equations (PDE) model certain wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves but that conserve energy. Probably the best known equation within the class of dispersive PDE is the nonlinear Schrodinger equation, which models a variety of physical waves, from signals in fiber optics to the macroscopic dynamics of the Bose-Einsten condensate. In the last fifteen years enormous progress has been made in settling fundamental questions on existence of solutions to the equation, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena that have been studied with sophisticated tools from nonlinear Fourier analysis, analytic number theory, and geometry. The principal investigator's main line of research has revolved around this aspect of mathematics for along time. More recently, a growing interest has been shown in incorporating nondeterministic points of view into the field of dispersive PDE. This has become one of the principal investigator's favorite lines of work in the last couple of years, and one that will be pursued in this project. Another new element in the project is the study of the "process" used by mathematical physicists to pass from a complex system of particles interacting with one another to a macroscopic wave function that is able to describe the most important features of the system as a whole. More precisely, the principal investigator will study the effective evolution equations arising as an appropriate limit of many body quantum dynamics.
From a mathematical point of view the types of problems that the principal investigator intends to explore in this project lie at the intersection of Fourier analysis, analytic number theory, numerical analysis, geometry, probability, and mathematical physics. Often it is difficult to "communicate" in so many different mathematical languages, but the time seems ripe to start such a conversation. The description of the problems that the principal investigator proposes in her research program should testify to this fact. Although she conducts her research mostly in the realm of pure mathematics, the solutions to the problems she considers have the potential to see very concrete and very diverse consequences in real life. For example, understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero are two very different phenomena in nature. On the other hand, they are both aspects of solutions to the same dispersive equation, the nonlinear Schrodinger equation that is one of the foci of this project, and they can be studied using the same mathematical language. In recent years, the world has witnessed incredible advances in technology. As scientists and engineers seek to take even greater strides in the technological arena, they are guided by experiments, but also by mathematical models, such as those provided by research on the Schrodinger equation. There is no doubt that solid predictions on the mathematical side will help target the experimental component in the most efficient and cost-effective way possible, thus saving resources as well as valuable time.
