Lattice, Trees and Group Actions

该计划的目标是促进对代数、群论和表示理论的理解，特别强调物理和几何的数学基础。Kac-Moody李代数是由物理学家首先发现的无限维代数。在20世纪70年代获得的这些代数的数学表征允许存在更广泛的一类Kac-Moody代数，即双曲代数。直到今天，数学中还没有对双曲代数的经典解释，也没有发现任何对它们的物理解释。然而，这些代数显示出显著的对称性，编码了数字和几何之间深刻而复杂的关系。这些双曲代数及其群是我们研究的对象。我们发现了有限域上kac - moody群格的自同构形式理论的开端。虽然Kac-Moody群没有明显的代数或算术结构，但我们的工作证明了它们与李群在正特征域上的相似性。在某种程度上，我们的方法是采用离散和组合方法来解决这个无限维环境中的问题。该课程的目标是提高对某些几何物体的代数和对称性的理解，特别强调物理和几何的数学基础。“Kac-Moody李代数”是数学家和物理学家都在研究的无限维空间，它最初是在物理学中作为“循环”代数被发现的，即从圆映射到有限维李代数。这些空间的数学表征，在20世纪70年代获得，允许存在更广泛的Kac-Moody代数，即双曲代数。直到今天，数学中还没有对双曲代数的经典解释，也没有发现任何对它们的物理解释。然而，这些代数显示出显著的对称性，编码了数字和几何之间深刻而复杂的关系。这些双曲代数及其对称性是我们研究的对象。本研究计划的一个重要主题是采用离散和组合方法来解决代数问题。提议者正在组建一个由来自不同国家的代数、几何、物理和组合学研究人员组成的团队，其中包括几名女性，以便为这个研究团队带来丰富多样的观点。

Abstract for award DMS-0401107 of CarboneThe objective of this program is to advance understanding inalgebra, group theory and representation theory withparticular emphasis on the mathematics underlying physicsand geometry. Kac-Moody Lie algebras are infinite dimensionalalgebras that were first discovered by physicists. A mathematicalcharacterization of these algebras obtained in the 1970's allowedfor the existence of a wider class of Kac-Moody algebras, namelyhyperbolic algebras. To this day, no classical interpretation ofhyperbolic algebras is known in mathematics, nor has any physicalinterpretation of them been discovered. Yet these algebrasdisplay remarkable symmetry properties which encode deepand intricate relationships between numbers and geometry.These hyperbolic algebras and their groups are the objectsof our study. We have discovered the beginnings ofa theory of automorphic forms for lattices in Kac-Moodygroups over finite fields. Although Kac-Moody groupshave no obvious algebraic or arithmetic structure, ourwork demonstrates substantial analogies with Lie groupsover fields of positive characteristic. Our approach, in part,has been to adapt discrete and combinatorial methods inorder to solve problems in this infinite dimensional setting.The objective of this program is to advance understanding inalgebra and symmetries of certain geometric objects with aparticular emphasis on the mathematics underlying physicsand geometry. "Kac-Moody Lie algebras" are infinite dimensionalspaces that are studied both by mathematicians and physicists,having first been discovered in physics as algebras of "loops",that is, maps from the circle into finite dimensional Liealgebras. A mathematical characterization of these spaces,obtained in the 1970's, allowed for the existence of a widerclass of Kac-Moody algebras, namely hyperbolic algebras.To this day, no classical interpretation of hyperbolicalgebras is known in mathematics, nor has any physicalinterpretation of them been discovered. Yet these algebrasdisplay remarkable symmetry properties which encode deepand intricate relationships between numbers and geometry.These hyperbolic algebras and their symmetries are theobjects of our study. A strong theme in this research programis to adapt discrete and combinatorial methods to solve problemsin algebra. The proposer is developing a team of researchersin algebra, geometry, physics and combinatorics from a widevariety of countries, and including several women, in orderto bring a wealth of diverse viewpoints to this research team.