Mathematical Analysis and its Applications

克雷格博士的研究涉及对偏微分方程解的数学研究，这些方程描述了物理现象中的波动演化，包括海浪、流体流动、量子力学和大尺度时空特征，涵盖了大尺度的光谱。他的研究项目包括波浪传播和海洋动力学问题，以及哈密顿力学技术及其在涡旋动力学和其他非线性波浪传播问题中的应用。他还将寻求与其他物理科学家在涉及波现象的各种问题上的合作。数学描述的本质是具有普适性，这解释了描述在如此多的情况和尺度下的有效性。克雷格博士的研究是理论性的，涉及数学思想和概念。然而，这些发现确实会导致其他学科的进步，其中的进步可以导致更有效的数值模拟或对实验的概念性理解。

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.