DMS-9801374 Lihe Wang This proposal is directed in the study of regularity theory of bi-harmonic maps, level surfaces of solutions of partial differential equations, problems of geometric flows and degenerate equations. Regularity of weakly bi-harmonic maps are established. Bi-harmonic maps are expected to play an important role in the theory of four manifolds. The level surfaces are the central subject of study in the theories of phase transitions, geometric evolutions and free boundary problems. Some special level surfaces appear as the boundary of crystals and the wave fronts of shock waves in many practical problems. We plan to develop some new ideas which help to understand the nature, stability and regularity of these level surfaces. The mathematical existence of flow of crystals and the regularity of the boundary of the crystals are also established. Since Newton discovered the second law of physics, progress in mathematical physics has relied on the partial differential equations, from fluid mechanics, relativity to quantum mechanics. Although partial differential equations describe nature quite precisely, they have many challenges. The major difficulty is due to their nonlinearity. Another big source of the complications comes from the singularities of some of the physical quantities. Physically these came from a phenomenon like that of the formation of a black hole or other physical phenomena such as explosion, phase transition or even super-conductivity. We try to understand these singularities by studying the structure and the configuration of the singularity set. A class of this kind of problems comes from crystal growth and free boundary problems. The motion of the boundary of the crystal is the geometric flow problem. Here the boundary of the crystal, such as in the case of melting ice, is a level surface of the temperature function. The whole dynamical process of the crystal growth is studied from This point of view.