Application of Galois cohomology to infinite dimensional Lie theory

李理论起源于19世纪末著名的挪威数学家索夫斯·李，它已经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影响。副产物是两个中心内在相关的对象，即李群和李代数。群（和代数）存在于所有数学中。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（后者是指在某些对称性下属性的持久性）。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。著名的公式e=mc2来自于一个简单的代数运算，如果我们假设物理的基本定律在平移和旋转下是不变的。所讨论的李群是仿射群和正交群的例子。这个思路非常丰富。例如，目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，但是用一种更容易操作的语言。李代数的表示，例如，对应于某些物理理论中的基本粒子。***在几何领域，李群和代数通常被视为“连续变化”的对象。由于数学上的原因，基于数论的应用，从20世纪50年代开始，几何学和李论的新“代数”版本得到了发展。这是代数几何和代数群的诞生，并与A. Grothendieck和他的合作者在60年代末发展的方案理论和约化群方案相结合。如果没有格罗腾迪克革命性的几何“语言”，过去三十年中确立的一些最深刻的数学结果（包括A.怀尔斯对费马大定理的证明）是不可能实现的。我的工作集中在试图发现格罗滕迪克的创作和无限维李理论之间的联系。“无限维度”部分自然地出现在弦理论中。从严格的数学角度来看，这也很重要。这为李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成果。最近发现的与其他数学领域的新联系表明，所提出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.**************