Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

Deift将继续研究强迫晶格振动、聚焦NLS方程的半经典极限、随机矩阵理论、可积模型和旋转黑洞。黎曼-希尔伯特技术将用于评估随机矩阵理论中出现的概率分布的渐近性，研究可积量子场论和统计力学模型中出现的渐近性问题，以及研究两个旋转黑洞之间的相互作用力，当洞的分离趋于无穷大时。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

PI: Deift DMS-9500867 Deift will continue his investigations into forced lattice vibrations, semi-classical limit for the focusing NLS equation, random matrix theory, integrable models, and rotating black holes. Riemann-Hilbert techniques will be used to evaluate the asymptotics of probability distributions that arise in random matrix theory, to investigate asymptotic questions that arise in the theory of integrable quantum field theories and statistical mechanical models, and to study the force of interaction between two rotating black holes in equilibrium as the separation of the holes goes to infinity. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.