Nonlinear Geometric Partial Differential Equations: Entire Solutions and Regularity

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Some of the most important problems in mathematics and physics are related to the understanding of singularities. Singularities can range anywhere from black holes in astrophysics to turbulence in fluid mechanics to the accumulation of cancer cells in biomedical research. Such physical phenomena are often described by differential equations that involve time and space. Studying the qualitative behavior of the solutions of these equations frequently deepens one's understanding of the related physical problems. To study a singularity of a solution one uses a so-called blow-up procedure that allows one to focus attention near the singularity and to exploit the scaling properties that the differential equation enjoys. Because of the change in the scaling of space and time, this process leads to a new solution that is defined for all space and time, in other words to a "global solution." The classification of global solutions, when possible, sheds new insight into the singularity and thus into the underlying physical phenomenon. This project addresses the questions of existence, uniqueness, and qualitative behavior of global solutions to nonlinear geometric elliptic and parabolic partial differential equations. Emphasis is given to the classification of ancient solutions, the construction of new ancient solutions from the gluing of solitons, and the study of fully nonlinear extrinsic geometric flows in the complete noncompact case. The interplay between analytical and geometric techniques will be a crucial factor in carrying out the research. The project links a wide range of active fields of mathematics, including nonlinear partial differential equations, differential geometry, and classical analysis. The principal investigator also intends to seek applications of the mathematical results to other disciplines such as quantum field theory and image processing. Results will be disseminated to the research community at various meetings and by publication of research articles. Special emphasis will be given to the training of Ph.D. students.