Hyperbolic Kac-Moody groups and algebras

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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.