Classification of Finite Simple Groups. Groups of even type of medium size.

有限简单群的分类（CFSG）被广泛认为是现代数学的主要成果之一。“这个证明有数千页长，涉及许多研究人员，可能是数学定理中最长、最复杂的证明”R.Guralnick。分类的最初证明是在2004年完成的，由M.Aschbacher和S.Smith的不朽著作《类群的分类》（The Classification of Quasithin Groups）完成，在此之前，100多名数学家花了几十年的时间在超过2万页的期刊上发表了数百篇文章。这项工作是数学界令人难以置信的集体努力。分类定理用于群论和数论、代数和算术几何、逻辑学和其他数学领域。由于作者、假设、假设和技术的异质性，已经写了一些书来解释CFSG的原始证明是如何构建的。即使有了这样的帮助，阅读完整的证明仍然是一项艰巨的任务。这一困难在早期就已经预料到，上个世纪就开始呼吁对CFSG进行新的统一的完整证明。这个由d.g orenstein, r.l ons和r.s oromon发起的项目的目的是产生一个新的“第二代”分类定理的统一证明。它的策略与最初的大不相同。第二代校样的预计长度约为5,000页，预计将由医疗辅助队出版超过12卷。目前，第1卷至第8卷已出版，第9卷已提交出版。正如R. Solomon在他的进度报告中所解释的那样，仍然存在两种情况：所谓的“p-唯一性情况”和“e(G)=3”情况。该系列的第10卷和第11卷致力于R. Lyons， R. Solomon和G. Stroth合作努力的“$p$-唯一性案例”。我参与第二代项目的主要目标是与R.Lyons， R.Solomon和Ch.Parker合作，产生有限单群分类新证明的一部分。我是第9卷的合著者，本提案的目的是寻求支持，以便在最后的第12卷中关注一个关键问题。

The Classification of Finite Simple Groups (CFSG) is widely acknowledged to be one of the major results in modern Mathematics. "The proof is thousands of pages long , involves many researchers and is likely the longest and most complicated proof of a theorem in Mathematics" R.Guralnick. The original proof of Classification was completed in 2004 with the monumental work of M.Aschbacher and S.Smith "The Classification of Quasithin Groups" after decades of work of more than 100 mathematicians in hundreds of articles spread over 20,000 journal pages. The work was an incredible collective effort of mathematical community. The Classification Theorem is used in group and number theory, algebraic and arithmetic geometry, logic, and other areas of mathematics.Because of the heterogeneity of authors, assumptions, hypotheses and techniques, books have been written to explain how the original proof of CFSG was constructed. Even with this help, the task of reading the complete proof would be daunting. This difficulty was already anticipated early on, and calls for a new unified complete proof of the CFSG started last century.The aim of the project initiated by D. Gorenstein, R. Lyons and R. Solomon is to produce a new "Generation-2" unified proof of the Classification Theorem. Its strategy differs considerably from the original one. The expected length of the Generation-2 proof is of about 5,000 pages and is projected to span over 12 volumes published by the AMS. At this moment Volumes 1 through 8 are published, and Volume 9 has been submitted for publication.As explained by R. Solomon in his progress report, two cases remain: the so-called "p-Uniqueness case" and the case "e(G)=3".Volumes 10 and 11 of the series are dedicated to the "$p$-Uniqueness case" in a collaborative effort of R. Lyons, R. Solomon and G. Stroth.The main goal of my involvement in the Generation-2 project isto produce a part of a new proof of Classification ofFinite Simple Groups in collaboration with R.Lyons, R.Solomon and Ch.Parker. I am a co-author in Volume 9, and the aim of this proposal is to ask for support to focus on a key problem in the final 12th Volume.