Classification of Finite Simple Groups. Second Generation Proof.

有限单群的分类被广泛认为是现代数学的主要成果之一。在20世纪80年代初，它的证明被宣布成功完成。然而，实际上，直到2004年，最初的证明才随着M.Aschbacherand S.Smith的巨著《类星体群的分类》而结束。有限简单群分类的原始证明大约有15000页，分布在500多篇独立的文章中，其中许多未发表，由100多位数学家撰写。尽管它令人印象深刻，但它立即使人怀疑是否有可能产生一个新的长度可能更短的有限简单群分类的统一完全证明。这个问题的答案成为了D.Gorenstein、r.l ons和r.s oromon发起的一个项目，其目标是：为分类定理提供新的第二代统一证明。它的战略与原来的战略有很大不同，并将在提案中加以概述。新一代第二证明的预计长度约为5000页，将在美国数学学会专著系列的11卷中发表。它们在续集中被称为gls系列。gls丛书的第一卷至第六卷已经出版。我在这个提案中讨论的项目是对gls系列第8卷的贡献，我一直在与R. Lyons和R. Solomon一起工作。分类定理的第二代证明的优点和最有趣的一点是，它将第一次给出零星单群的统一特征，这些神秘的物体的生活直到现在还没有一个统一的解释。特别是，第8卷解释和描述了26个零星的简单群中的11个，其中最大的一个是怪物群。因此，本拨款申请的主要目的是确保支持将允许成功发展和可能完成现代数学主要成果之一的证明的某些部分：有限单群分类。

The Classification of Finite Simple Groups is widely acknowledged to be one of the major results in modern Mathematics.The successful completion of its proof was announced in the early1980's. However, in reality, it was only in 2004 that the originalproof has come to an end with the monumental work of M.Aschbacherand S.Smith: The Classification of Quasithin Groups .The original proof of the Classification of Finite Simple Groupsoccupies somewhere around 15,000 journal pages spread across morethan 500 separate articles, many unpublished, written by more than100 mathematicians. As impressive as it is, it immediately makesone wonder whether it would be possible to produce a new unifiedcomplete proof of Classification of Finite Simple Groups possiblyshorter in length.An answer to that question became a project started by D.Gorenstein, R. Lyons and R. Solomon, the goal of it being: toproduce a new Generation 2 unified proof of the ClassificationTheorem. Its strategy considerably differs from the original oneand will be outlined in the proposal. The expected length of thenew Generation 2 Proof is of about 5,000 pages to be publishedin 11 volumes of the American Mathematical Society MonographSeries. They be referred to in the sequel as the GLS-Series.Volumes 1 through 6 of the GLS-Series have already been published.The project which I am discussing in this proposal and have beenworking on together with R. Lyons and R. Solomon is a contributionto Volume 8 of the GLS-Series. One of the advantages and mostinteresting points of the Generation 2 Proof of the ClassificationTheorem, is that it will give the first unified characterizationof Sporadic Simple Groups, the mysterious objects, whose life did not have a unified explanation untill now. In particular, Volume8 explains and characterizes 11 out of 26 sporadic simple groupsamong which is the largest one, the Monster Group.Thus the main purpose of this Grant Application is to ensure asupport which will allow the successful development and possiblecompletion of some parts of the proof of what is known as one ofthe major results of Modern Mathematics: Classification of FiniteSimple Groups.