Lattice, Trees and Group Actions

摘要奖DMS-0401107碳这个计划的目的是促进理解代数，群论和表示论，特别强调数学基础的物理和几何。Kac-Moody李代数是最早由物理学家发现的无限维代数。在20世纪70年代得到的这些代数的一个代数特征允许存在更广泛的一类Kac-Moody代数，即双曲代数。直到今天，在数学中还没有发现双曲代数的经典解释，也没有发现它们的物理解释。然而，这些代数显示出显著的对称性，这些对称性编码了数与几何之间深刻而复杂的关系，这些双曲代数和它们的群是我们研究的对象。我们发现了有限域上Kac-Moody群格的自守形式理论的开端。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与正特征域上的李群有实质的相似性。我们的方法部分是采用离散和组合方法来解决无限维环境中的问题。该计划的目标是促进对某些几何对象的代数和对称性的理解，特别强调物理和几何基础的数学。“Kac-Moody李代数”是数学家和物理学家都在研究的无限维空间，在物理学中首先被发现为“循环”代数，即从圆到有限维李代数的映射。在20世纪70年代得到的这些空间的数学特征使得更广泛的一类Kac-Moody代数，即双曲代数的存在成为可能。直到今天，在数学中还没有发现双曲代数的经典解释，也没有发现它们的任何物理解释。然而，这些代数显示出显著的对称性，这些对称性编码了数与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。一个强有力的主题，在这个研究计划是适应离散和组合的方法来解决问题的代数。提议者正在发展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，其中包括几名妇女，以便为这个研究团队带来丰富的不同观点。

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.