Abstract Chen Chen will investigate problems in two areas: chaotic behavior of the Lorenz system and the interfacial phenomena in material science. For the first area Chen seeks to obtain a complete bifurcation diagram of the chaotic behaviors of the system for all the parameter values and to show the existence of a conjectured Lorenz attractor. The main tools to be used will be shooting arguments, homotopy theory, bifurcation theory, and chaos theory on maps. For the second area the goal is to study certain free boundary problems which describe geometric motions of hypersurfaces, to analyze the asymptotic behavior of certain evolution equations, and to connect the solutions of free boundary problems with the asymptotic behaviors of the evolution equations. The Lorenz system was one of the very first examples that stimulated the formation and development of the theory of chaos and later on became a typical example in many text books of dynamical systems and/or chaos theory. Nevertheless, except for numerous numerical experiments, there are few rigorous analyses for this system. The object of this project is to provide such a rigorous foundation. The related study of interfacial dynamics is of particular importance in material science.
