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

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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.