Enshrining Finite Simple Groups

PI将处理有限单群、格和顶点算子代数(VOA)中的问题。VOAS的工作是与董崇英联合进行的。它的主要目标是为有限的简单群体建立月光般的VoA(就像80年代中期对怪物所做的那样)。晶格的构造和分析支持这一目标，但由于新的技术和应用，最近关于晶格的工作有了自己的生命。(1)国际和平倡议计划通过减少假设，加强最近关于重要的月光美国之音的独特性的联合工作。(2)PI计划给出新的零星团体和供奉它们的VoA的结构。这将涉及对现有月光美国之音理论的修订和对特定零星群体的适应，以及与更经典的格型VoA的其他工作。其结果将是大多数有限单群的新的和一致的设置。(3)PI最近关于格的工作主要集中在Barnes-Wall级数的衍生上。国际和平研究所将继续进行这项研究。PI将用一系列有限群作为自同构群来构造新的格系列。这些系列和衍生产品中的某些将在第(2)部分中使用。其中一些晶格具有相对较高的最小范数(并且可能是极端的)，因此PI将尝试解决这些范数。其中一些格子的自同构群将被确定。独特性理论将会得到发展。(4)PI计划进一步研究低阶VoA的自同构群以及非结合代数与VoA之间的联系。国际数学联合会希望这项提议将有助于将零星的群体融入传统的主流数学。零星单群是有限单群，它们自然不是无穷级数(经典群、交错群)的一部分。带有格的程序将涉及有限域上的经典群的格无穷级数与与零星群相关的格级数联系起来。该程序将与VOA理论联系起来，将有限群、一般代数群和无限维李理论联系起来。格子、Voa和单群与数学和数学物理的许多部分相联系。

The PI will work on problems in finite simple groups, lattices and vertex operator algebras (VOAs). Work on VOAs is joint with Chongying Dong. The main goal is to build moonshine-like VOAs for finite simple groups (as done for the monster in the mid 80s). Lattice constructions and analyses support this goal but recent work on lattices has a life of its own due to new techniques and applications. (1) The PI plans to strengthen recent joint work on uniqueness for the important moonshine VOA by reducing hypotheses. (2) The PI plans to give new constructions of sporadic groups and VOAs which enshrine them. This will involve revisions of existing moonshine VOA theories and adaptions to particular sporadic groups, and other work with the more classic lattice type VOAs. The result would be a new and uniform setting of most finite simple groups. (3) The PI's recent work on lattices has concentrated on spinoffs of the Barnes-Wall series. The PI will continue this study. The PI will build new series of lattices with series of finite groups as automorphism groups. Certain of these series and spinoffs will be used in part (2). Some of these lattices have relatively high minimum norms (and could be extremal), so the PI will try to settle those norms. Automorphism groups of some of these lattices will be determined. Uniqueness theories will be developed. (4) The PI plans to do more work on automorphism groups of low rank VOAs and on the connections between nonassociative algebras and VOAs. The PI hopes this proposal will help integrate sporadic groups into traditionally mainstream mathematics. Sporadic simple groups are finite simple groups which are not naturally part of the infinite series (classical groups, alternating groups). The program with lattices links certain infinite series of lattices involving classical groups over finite fields with those associated to sporadic groups. The program with VOA theory will link finite groups, general algebraic groups and infinite dimensional Lie theory. Lattices, VOAs and simple groups are connected to many parts of mathematics and mathematical physics.