Michael Aschbacher of the California Institute of Technology discussed the proof of the classification of finite simple groups, a massive undertaking that has stretched over decades. For many years the status of this long, complicated, and diffuse proof has been unclear. But now the proof will finally be completed with the appearance a 1200-page manuscript by Aschbacher and Steve Smith, to be published in two volumes by the AMS.
Finite simple groups have a special significance in mathematics as the "building blocks" for all finite groups, in much the same way that the prime numbers are the "building blocks" for the integers. The problem of classifying all finite simple groups comes down to proving that they fall into four distinct classes: groups of prime order, alternating groups, groups of Lie type, and sporadic groups (of which there are 26). Around 1980, Daniel Gorenstein and Richard Lyons began an effort to write down clearly and carefully the proof of this classification; Ronald Solomon soon joined them in the effort. After Gorenstein's untimely death in 1992, Lyons and Solomon continued the work in a series of volumes also published by the AMS. The 1200-page work by Aschbacher and Smith fills a gap that arose in the original proof, concerning what are known as quasi-thin groups. Aschbacher feels strongly that the long and complicated proof is unsatisfactory, as it relies on a large number of papers scattered throughout the literature. The beauty of the classification---together with its applications in such areas as group theory, combinatorics, number theory, and algebraic geometry---make it important that a full proof be written down carefully in one place.
--- Allyn Jackson, Deputy Editor of The Notices