Applications of Galois cohomology to infinite dimensional Lie theory

我的工作是在李理论，所谓的命名后，挪威数学家索菲斯李（发音李），在19世纪后期，开始研究这一特定领域的数学。李理论的核心是对称性的概念，包括离散和连续。 大自然为我们提供了大量关于离散对称性的证据，例如雪花、分子和晶体所表现出的对称性。这些对称性的存在不仅具有数学意义，而且深刻地反映了物体本身的物理特性（例如，分子结合在一起的强度）。 连续对称性是李约瑟思想的核心，它在现代物理学中发挥了并将继续发挥重要作用，有时为特定问题或理论提供了决定性的见解。李理论的不同部分是作为自然定律中假设的对称性的标志出现的;例如经典力学中的空间旋转和平移，量子力学中的海森堡群，狭义相对论中的洛伦兹群。在这些例子中，有一条重要的共同哲学线索：一旦被给定的对称性所限制，自然定律往往会尽可能简单。在物理学中，这一原理是如此普遍，以至于当前的一些统一理论（如超弦）的定律都是以这种方式导出（或证明）的。 在硬科学之外，我们也被对称性所包围，正如贝瑟尔的绘画、巴赫的赋格和达芬奇的研究所表明的那样。因此，可以毫不夸张地说，柏拉图意义上的对称性是我们本性的一部分。谎言理论为我们进一步理解这种关系提供了一个强有力的工具。

My work is in Lie theory, so-named after the Norwegian mathematician Sophus Lie (pronounced Lee) who, in the late 19th century, began the study of this particular field of mathematics. At the heart of Lie theory is the idea of symmetry, both discrete and continuous. Nature provide us with ample evidence of discrete symmetries, such as those exhibited by snow flakes, molecules and crystals. The existence of these symmetries is not only of mathematical interest, but reflect deeply on the physical characteristics of the objects themselves (how strongly can molecules bind together, for example). Continuous symmetries, which were at the heart of Lie's ideas, have played and continue to play a fundamental role in modern physics, providing at times decisive insight into a particular problem or theory. Different parts of Lie theory arise as signs of the postulated symmetry in the laws of nature; for instance rotations and translations in space for classical mechanics, the Heisenberg group in quantum mechanics, and the Lorentz group in the case of special relativity. There is an important common philosophical thread weaving through these examples: The laws of nature tend to be as simple as possible once they are circumscribed by a given group of symmetries. In the case of physics this principle is so pervasive that the laws of some of the current unifying theories (like superstrings) are derived (or conjectured) in this way. Outside the hard sciences we are also surrounded by symmetry, as Escher's drawings, Bach's fugues and Da Vinci's studies illustrate. It is thus not an exaggeration to say that symmetry--in the Platonic sense--is part of our very nature. Lie theory affords us a powerful tool for furthering our understanding of this relationship.