Application of Galois cohomology to infinite dimensional Lie theory

李理论起源于19世纪末的挪威著名数学家索菲斯·李，现已发展成为一门重要的数学学科，在许多其他数学领域都有深远的触角。副产品是两个中心本质上相关的对象，即李群和李代数。 群(和代数)存在于所有的数学中。它们最重要的应用之一是检测/描述对象的对称性和不变性(后者是某个属性在一定对称性下的持久性)。对称对许多科学领域的重要性是至高无上的。这方面的典型例子可以在爱因斯坦的狭义相对论中找到。如果假设物理学的基本定律在平移和旋转下必须是不变的，那么著名的公式e=mc2就来自一个简单的代数运算。所讨论的李群是仿射群和正交群的例子。这一思路极其丰富。例如，当前粒子物理学中的一些奇异理论(例如，任何超弦理论)通过先验地假设理论必须尊重某些对称性来推导公式。李代数编码了群的大部分信息，但使用了一种更容易操作的语言。例如，李代数的表示对应于某些物理理论中的基本粒子。 在几何学领域，李群和代数通常被认为是“连续变化的”对象。由于数学上的原因，从1950年的S开始，基于数论的应用，几何学和李理论的新的“代数”版本被发展起来。这就是代数几何和代数群的诞生，它后来与格罗森迪克和他的合作者在60年代末S发展的图式和约化群方案的理论相融合。过去30年来建立的一些最深刻的数学结果(包括A·威尔斯对费马大定理的证明)，如果没有格罗森迪克革命性的几何语言，是不可能的。 我的工作重点是试图发现格罗森迪克的创作和无限维谎言理论之间的联系。“无限维”部分在弦理论中自然出现。从严格的数学观点来看，它也很重要。这为谎言理论的研究提供了强大的新机制，并导致了许多基本结果的产生。最近发现的与其他数学领域的新联系表明，拟议的研究方法将继续产生最高科学标准的结果。

Lie theory, which owes its origins to the famous Norwegian mathematician Sophus Lie at the end of the 19th century, has evolved into a major mathematical subject, with far reaching tentacles in many other mathematical areas. A by-product is two central intrinsically related objects, viz., Lie groups and Lie algebras. Groups (and algebras) are present in all of mathematics. One of their most important applications is to detect/describe symmetry and invariance of objects (the latter being the permanence of a property under certain symmetries). The importance of symmetry to many areas of science is paramount. The quintessential examples of this can be found in Einstein's theory of special relativity. The famous formula e=mc2 follows from a simple algebraic manipulation if one assumes that the basic laws of physics have to be invariant under translations and rotations. The Lie groups in question are examples of affine and orthogonal groups. This line of thought is extremely fertile. For example, some of the current exotic theories in particle physics (e.g., any of the superstring theories) derive formulas by assuming a priori that the theory must respect certain symmetries. The Lie algebras encode much of the information of the groups but in an easier language to manipulate. The representations of the Lie algebras, for examples, correspond to elementary particles in certain physical theories. In the area of Geometry, Lie groups and algebras are typically regarded as “continuously varying” objects. For mathematical reasons, based on applications to number theory, starting in the 1950's, new “algebraic” versions of geometry and Lie theory were developed. This was the birth of algebraic geometry and algebraic groups, which went on to amalgamate with the theory of schemes and reductive group schemes, as developed by A. Grothendieck and his collaborators in the late 60's. Some of the deepest mathematical results established over the last three decades (including A. Wiles's proof of Fermat's last theorem) could not have been possible without Grothendiecks's revolutionary “language” of geometry. My work centers in trying to discover connections between Grothendieck's creations and infinite dimensional Lie theory. The “infinite dimensional” part appears naturally in string theory. It is also important from a strictly mathematical point of view. This has furnished powerful new machinery to the study of Lie theory and that led to fruition a number of fundamental results. Recently discovered new connections to other areas of mathematics indicate that the proposed research approach will continue to produce results of the highest scientific standards.