Enshrining Finite Simple Groups

PI将研究有限单群、格和顶点算子代数（VOAs）中的问题。VOAs的工作是与董崇英共同完成的。我们的主要目标是为有限的简单群体创造类似月光酒的voa（游戏邦注：就像80年代中期的怪物那样）。晶格结构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) PI计划通过减少假设来加强最近对重要的私酿之音的独特性的联合研究。(2) PI计划新建一些零星的团体和纪念他们的voa。这将涉及对现有的私酿之音理论的修订，并对特定的零星群体进行调整，以及对更经典的点阵型VOAs进行其他工作。结果将是大多数有限单群的一个新的统一的集合。PI最近对格子的研究主要集中在Barnes-Wall系列的衍生品上。PI将继续这项研究。PI将用有限群的序列作为自同构群来构建新的格序列。这些系列和衍生产品中的某些将在第(2)部分中使用。其中一些格具有相对较高的最小规范（并且可能是极端的），因此PI将尝试解决这些规范。其中一些格的自同构群将被确定。独特性理论将得到发展。(4) PI计划在低秩voa的自同构群以及非结合代数与voa之间的联系上做更多的工作。PI希望这一提议将有助于将零星的群体整合到传统的主流数学中。偶发单群是有限单群，不属于无限级数（经典群，交替群）。带格的程序将有限域上涉及经典群的无限格序列与零星群的无限格序列联系起来。该计划与VOA理论将连接有限群，一般代数群和无限维李论。晶格、voa和简单群与数学和数学物理的许多部分有关。

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.