About a month ago, Slashdot had linked to a completely opaque article about something called the Serre conjecture. The subsequent thread showed Slashdot at its worst, as people desperately typed “Serre conjecture” into Google to find links to stick in a comment to earn some undeserved karma. Unfortunately, Serre was an influential and productive mathematicians, so there are lots of Serre conjectures. One particularly unfortunate soul hit upon a link to the Quillen-Suslin theorem, which was conjectured by Serre in the 1950s but proven in 1976, and complained “do people not do research any more to see if their work has already been done?”
The Serre Conjecture in question is a completely separate conjecture in Galois theory. Many questions in number theory can be reduced to studying the Galois group of the algebraic closure of the rationals over the rationals. This group is gigantic and extremely complicated, so mathematicians try to understand it in pieces, but the pieces themselves are hard to come by. Serre conjectured one approach to understand certain small pieces. The paper in question, by Chandrashekhar Khare, proves a part of that conjecture. So Khare proved a part of a conjecture that only provides a small part of the solution of the general problem, but even that small part has remained unsolved for 18 years. So this is a hard problem we’re talking about. The truly macho can download the paper itself at ArXiv.