Mathematical Analysis and its Applications

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Dr. Craig's research involves mathematical studies of the solutions of partial differential equations that describe wave evolution in physical phenomena, including ocean waves, fluid flows, quantum mechanics, and large scale features of space-time, covering a large spectrum of scales. His research program in this proposal includes problems of wave propagation and ocean dynamics, and techniques of Hamiltonian mechanics and their applications to vortex dynamics and other nonlinear wave propagation problems. He also will pursue collaborations with other physical scientists on a variety of problems involving wave phenomena. It is the nature of mathematical descriptions to possess properties of universality, which explains the effectiveness of descriptions in this large variety of situations and scales. Dr. Craig's research is theoretical, involving mathematical ideas and concepts. However these discoveries do lead to progress in other disciplines, where advances can lead to more efficient numerical simulations or to a conceptual understanding of experiments.