Recently Isabel Lugo asked about problems that are hard for intermediate values of some parameter, and in discussing the question I got to thinking about exceptional structures in mathematics such as the sporadic groups. In 2006 David Corfield asked about how “natural” the sporadic simple groups are at the n-Category cafe. In that discussion and more generally there seem to be approximately two extremes in perspective:
- Exceptional structures represent a lack of room for asymptotic behavior to occur; thus they are distractions from the “generic” case. This seems to be the case for certain exceptional isomorphisms; there are only so many groups of a particular small order. It also seems to be a good way to think about objects that behave fine in characteristic zero or high characteristic but behave badly in low characteristic, characteristic
usually being the worst offender.
- Exceptional structures represent the deepest part of a theory, and the exceptional structures in different fields are often related; thus understanding exceptional structures is crucial. This seems to be the case for the octonions, which can be thought of as an underlying cause of Bott periodicity. It also seems to be a good way to think about objects related to the number
; John Baez tells a great story about connections between the Leech lattice, the Dedekind eta function, string theory, and elliptic curves all centered around this mysterious number.
usually being the worst offender.
So what do you think? Are exceptional structures accidents or miracles? (Or, as a third option: am I failing to distinguish carefully enough between interesting and uninteresting exceptional structures?)