Applications of Galois cohomology to infinite dimensional Lie theory

伽罗瓦上同调在无限维李理论中的应用

• 批准号: 9343-2011
• 负责人: Pianzola, Arturo
• 金额: $ 2.55万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568663
• 关键词: Applications; Galois; cohomology; infinite; dimensional

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.

我的工作是李理论，以挪威数学家 Sophus Lie（发音为 Lee）的名字命名，他在 19 世纪末开始研究这一特殊的数学领域。李理论的核心是对称性的概念，包括离散的和连续的。 大自然为我们提供了充分的离散对称性证据，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有多强）。 连续对称性是李思想的核心，在现代物理学中已经并将继续发挥基础性作用，有时为特定问题或理论提供决定性的见解。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而出现的。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以及狭义相对论中的洛伦兹群。这些例子中存在着一条重要的共同哲学线索：一旦自然法则受到一组给定的对称性的限制，它们就会变得尽可能简单。就物理学而言，这一原理是如此普遍，以至于当前一些统一理论（如超弦）的定律都是以这种方式推导（或推测）的。 在硬科学之外，我们也被对称性所包围，正如埃舍尔的绘画、巴赫的赋格曲和达芬奇的研究所表明的那样。因此，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。谎言理论为我们提供了一个强大的工具来加深我们对这种关系的理解。