Classification of Finite Simple Groups. Second Generation Proof.

有限单群的分类是现代数学中公认的重要结果之一，它的证明在20世纪80年代初就已成功完成。然而，实际上，直到2004年，最初的证明才随着M. Aschbacher和S.Smith的不朽著作《拟薄群的分类》（The Classification of Quasithin Groups）而结束。有限单群分类的最初证明大约有15，000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统一完整的证明有限单群的分类可能在长度上更短。里昂和R.所罗门，它的目标是：产生一个新的第二代统一证明的分类定理。它的战略与最初的战略有很大不同，将在提案中概述。新的第二代证明的预期长度约为5,000页，将被收录在美国数学学会专著系列的11卷中。GLS系列的第一至第六部分已经出版，我在这个建议中讨论的项目是与R.里昂和R.所罗门是对GLS系列第8卷的贡献。分类定理的第二代证明的一个优点和最有趣的一点是，它将首次统一地刻画散生单群这一至今尚无统一解释的神秘对象。特别是，第8卷解释和表征了26个零星的简单群中的11个，其中最大的是怪物群。因此，本资助申请的主要目的是确保支持，这将允许成功的发展和可能完成的证明的某些部分，这被称为现代数学的主要成果之一：简单群的分类。

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Finite Simple Groupsoccupies somewhere around 15,000 journal pages spread across morethan 500 separate articles, many unpublished, written by more than100 mathematicians.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.