Mathematical Sciences: Research in Modern Analysis Applicable to Fundamental Physical Theory

欧文·西格尔教授是一位极具创造力、涉及面广且极具影响力的数学科学家。他的开创性贡献跨越了50年的研究范围，从非交换积分理论、算子代数、表示理论、交换和非交换调和分析、李群，到量子物理学，以及最近的河外天文学。西格尔教授的研究仍然是最高水平的，他丰富的思想仍然是非常美丽的数学和对宇宙基本物理定律的基本见解的源泉。目前的提议在一个宏大的尺度上处理数学物理。它的目的是利用一系列复杂的数学思想，从泛函和谐波分析、几何、群表示、非线性偏微分方程、散射理论等方面对基本物理定律进行全面的研究。这项工作的核心是四维时空中的波动方程（和其他质量零方程），解在闵可夫斯基空间之外的扩展，以及非线性波动方程的量化。非线性抛物型偏微分方程在黎曼几何中的应用与经验基本粒子现象的建模密切相关。这是一个巨大的数学工程，对广阔的自然科学领域具有重大意义。

Professor Irving Segal has been an extraordinarily creative, broad, and influential mathematical scientist. His seminal contributions spread over five decades span a range of research from the theory of noncommutative integration, operator algebras, representation theory, commutative and noncommutative harmonic analysis, Lie groups, to quantum physics, and more recently extragalactic astronomy. Professor Segal's research is still of the highest calibre, and his fertile mind remains a font for both remarkably beautiful mathematics and foundational insights into the fundamental physical laws of the universe. The current proposal treats mathematical physics on a grand scale. Its aim is a comprehensive study of fundamental physical laws utilizing a range of sophisticated mathematical ideas drawn from functional and harmonic analysis, geometry, group representations, nonlinear partial differential equations, scattering theory, and more. Central to this work is the wave equation (and other mass zero equations) in four-dimensional space-time, the extension of solutions beyond Minkowski space, and the quantization of nonlinear wave equations. Applications to the Riemannian geometry of nonlinear parabolic partial differential equations go hand in hand with modelling of empirical elementary particle phenomena. This is an immense mathematical project of great significance to a vast expanse of natural science.