Mathematical Sciences: Invariant Variational Problems and Quantization

李群理论，以挪威人的荣誉命名 数学家Sophus Lie，一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性，李理论 对数学本身和理论产生了深远的影响 物理学，尤指量子力学，基本粒子 物理学和相对论 特别是，结构和 对称性是交织在一起的，正如基本功所说明的那样， 爱因斯坦和后来的杨-米尔斯，在其中取得了深刻的成功， 关于自然界基本力的非线性理论 在数学上，需要一个大家庭， 对称性 在另一个方向，数学家和 物理学家们，通常是独立的，已经开发了大量的 强大的技术，分类的线性实现， 对称性 这就是“表征理论”的主题。 朱克曼教授在1998年进行了开创性的创造性工作， 李群表示论的许多方面，他现在是 将他的研究成果应用于基础物理学。 十年前他 发明了新的方法来构造Lie的线性表示 他的方法从不同的，看似无关的 数学领域。 最近，朱克曼教授的努力 已经被引导去适应令人兴奋的新的 研究物理学中的“弦理论”。 这个理论的目标是 来统一描述自然界的所有力量 在 根据目前的提议，他将对线性和 弦理论中对称性的非线性表现，以及 其他更传统的物理理论。 这包括 发展一个新的基础，从古典到 量子物理理论，以及两者的兼容性 爱因斯坦的狭义相对论和广义相对论 对称性在经典水平上实现非线性， 在量子水平上是线性的。 对称性的这种双重作用是 他工作的基本方面。

The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics, elementary particle physics, and relativity. In particular, the structures and the symmetries are interwoven, as illustrated by the fundamental work of Einstein, and later Yang-Mills, in which profoundly successful nonlinear theories for the fundamental forces in nature have been guided mathematically by the need for a large family of symmetries. In a different direction, mathematicians and physicists, often independently, have developed a vast array of powerful techniques that classify the linear realizations of the symmetries. This is the subject of "representation theory". Professor Zuckerman has done pioneering creative work in many facets of representation theory of Lie groups, and he is now applying his work to fundamental physics. Ten years ago he invented new ways to construct linear representations of Lie groups, his methods drawing from diverse, seemingly unrelated areas of mathematics. Recently, Professor Zuckerman's efforts have been directed toward their adaptation to the exciting new study of "string theory" in physics. The goal of this theory is to provide a unified description of all the forces of nature. In the current proposal he will conduct research on both linear and nonlinear manifestations of symmetry in string theory, as well as other more conventional physical theories. This includes the development of a new basis for the passage from classical to quantum physical theories, and resulting compatibility of both theories with Einstein's special and general relativity. Symmetries are realized nonlinearly at the classical level and linearly at the quantum level. This dual role of symmetry is a fundamental aspect of his work.