Hyperbolic Kac-Moody groups and algebras

该项目的技术目标是(1)对“有趣的”Kac-Moody李代数的可能的根系进行分类，其中“有趣的”具有特定的技术含义；(2)计算出相应的Kac-Moody群的基本性质，特别是在整数环上，特别是在像E10和E11代数这样物理上重要的情况下；(3)加强目前猜测的对零星有限简单的“怪物”群在复双曲几何和代数几何中的作用的支持。更具体地说：为了理解一个特定复杂的奥比福尔德的基本群，已经被推测为怪物。计算机计算将在第一个项目中发挥很大作用，其他项目更具理论意义。这个项目最直接的更广泛的意义是第(2)部分的意义：大量物理学家正在研究称为E11(Z)的群作为称为M理论的物理理论的对称群的猜想作用。关于这一群体的有用的计算描述缺失，丽莎·卡伯恩和我的目标是提供一个。(还有许多其他事情，包括回答更纯粹的数学问题。)与任何分类项目一样，第(1)部分将发现一些“宝石”和一些不太有趣的对象。已知的宝石(如怪物李代数)非常深，在数学上具有深远的影响，所以我们自然想找到剩下的。第(3)部分更具投机性；许多巧合表明它会成功，如果成功了，它将在相距甚远的两个数学部分之间架起一座桥梁。第(2)部分明确的协作性质，以及与这些问题相关的问题对研究生的适用性，意味着该研究非常适合数学/物理学界其他人的更广泛的研究领域。

The technical goals of the project are (1) to classify the possible root systems for "interesting" Kac-Moody Lie algebras, where "interesting" has a specific technical meaning, (2) work out the basic properties of the corresponding Kac-Moody groups, especially over the ring of integers, especially in physically-important cases like the E10 and E11 algebras, (3) strengthen the currently-conjectural support for a role of the sporadic finite simple "Monster" group in complex hyperbolic geometry and algebraic geometry. More specifically: to understand the fundamental group of a particular complex orbifold, already conjectured to be the Monster. Computer calculation will play a large role in the first project, and the others are more theoretical.The most immediate broader significance of this project is that of part (2): a large number of physicists are studying the conjectural role of a group called E11(Z) as the symmetry group of a physical theory called M-theory. A useful-for-calculation description of this group is missing, and Lisa Carbone and I aim to provide one. (And many other things, including addressing questions more purely mathematical.) Like any classification project, part (1) will turn up a few "jewels" and also some not-quite-so-interesting objects. The jewels already known (like the monster Lie algebra) are very deep, with far-reaching influence in mathematics, so we naturally want to find the rest. Part (3) is more speculative; many coincidences suggest it will work out, and if it does then it will build a bridge between two parts of mathematics that have been far apart. The explicitly collaborative nature of part (2), and the suitability of problems related to these questions for graduate students, mean that the research fits well into a larger area of study by others in mathematics/physics community.