Prize awarded for largest mathematical proof

The largest proof in mathematics is colossal in every dimension – from the 100-plus people needed to crack it to its 15,000 pages of calculations. Now the man who helped complete a key missing piece of the proof has won a prize.

In early November, Michael Aschbacher, an innovator in the abstract field of group theory at the California Institute of Technology in Pasadena will receive the $75,000 Rolf Schock prize in mathematics from the Royal Swedish Academy of Sciences for his pivotal role in proving the Classification Theorem of Finite Groups, aka the Enormous Theorem.

If it were not for Aschbacher, the behemoth might still contain a gaping hole. In 2004, he and Stephen Smith of the University of Illinois in Chicago published a 1200-page guide through the last piece of the puzzle.

That 2004 tome brought together some of Aschbacher's early works, and completed the proof. His contributions to the overall proof were "absolutely monumental", says Ronald Solomon, a group theorist at Ohio State University in Columbus.

Elemental groups

The Enormous Theorem concerns groups, which in mathematics can refer to a collection of symmetries, such as the rotations of a square that produce the original shape. Some groups can be built from others but, rather like prime numbers or the chemical elements, "finite simple" groups are elemental.

There are an infinite number of finite simple groups but a finite number of families to which they belong. Mathematicians have been studying groups since the 19th century, but the Enormous Theorem wasn't proposed until around 1971, when mathematician Daniel Gorenstein of Rutgers University in New Jersey devised a plan to identify all the finite simple groups, divide them into families and prove that no others could exist.

Gorenstein and his hundreds of collaborators spent a decade working on the proof. By 1981, Gorenstein could see the light at the end of the tunnel, though a few hurdles remained. The proof remained incomplete until the 2004 publication by Aschbacher and Smith, which completed the proof. It identified all the families – and showed no others could exist.

Diabolically difficult

It also identified all the known "sporadic groups", 26 (or 27, according to some) outlying simple finite groups that get pooled as one family because they do not fit neatly into the other families.

Solomon estimates that only a few mathematicians in the world (including Aschbacher) understand the complete proof. It was a punishing read, saysMark Ronan, an honorary professor of mathematics at University College London. "Some of Aschbacher's proofs were just diabolically difficult," he adds.

Mathematicians cannot predict how the proof will influence the future of mathematics or the sciences. Ronan says that like many mathematical results, the applications may not surface any time soon. "Whatever it's telling us, we haven't yet found out," he says. "I'd be willing to bet a million dollars that it has an application, but there's no point in making the bet because I'll be dead before I can collect."