New Developments (since 1977)

Abstract

With certain conjectures in mathematics, the solution of the conjecture marked the end of a period of investigation, and workers in the field moved on to other directions of research. With other conjectures, the solution of a once-open problem would lead to analogues and extensions of the problem, and provide a fountainhead of ideas for further generalizations, or even formulation of significant new problems. Fortuitously, Serre’s Conjecture is of the latter kind. In fact, the continuing research on problems related to Serre’s Conjecture in the post Quillen-Suslin period (that is, 1977-present) turned out to be even more intensive and voluminous than the work done on the original conjecture between 1955 and 1976. In total, at least some six hundred papers have appeared after 1976 that, in one manner or another, impacted upon Serre’s Problem — by way of refinements, generalizations, adaptations, variations, or applications. The surprising level, quality, quantity, and breadth of this ensuing work is, in my opinion, truly a tribute to the genius of Quillen, Suslin, and Serre.