# Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind:

-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one believes the Classification is indeed complete)

-Classification of finte-dimensional semisimple Lie algebras with 4 infinite families and 5 exceptional examples

-Classification of (Simple, Formally Real) Jordan Algebras with 4 infinite families and 1 exceptional example

I'm sure there are other examples that non-algebraists would think of before these. All the examples I cited take the basic form of having several infinite families and some number of exceptional examples which do not fall into any of these families. Thus I am wondering:

Question: Does anyone know of examples of classifications of some mathematical objects such that the classification consists (A) only of infinite families or (B) only of a finite number of examples/ an infinite number of examples which do not seem to be closely related to one another (i.e. they do not "appear" to form any infinite families).

One example of case (B) that comes to mind would be finite dimensional division algebras over $\mathbb{R}$ of which there are 4. On the other hand, for case (B) I would like to rule out way too specific "classifications" such as "finite simple groups with an involution centralizer of such and such a form" since this is really a subclassification within the classification of FSG's. Although I am an algebraist, I would like to hear about examples from any branch of math, for comparison's sake.

(If anyone thinks of better tags for this, feel free to add or suggest them).

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Often the representations of some algebraic object that you care about can be described in some uniform way. Say the polynomial representations of $GL_n(\mathbb{C})$ or finite-dimensional $\mathfrak{sl}_2$ modules. – Sam Hopkins Jun 16 '14 at 20:05
– David Roberts Jun 17 '14 at 0:55

This feels like a somewhat silly example, but what about the classification of two (real) dimensional manifolds? They are two families of these, the orientable and non-orientable families classified by their genus/Euler characteristic.

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I should add that this is an example of (A). – Simon Rose Mar 24 '11 at 18:53
Wow, I guess I totally overlooked such an obvious example from undergrad/grad topology classes. – ARupinski Mar 24 '11 at 18:53
@Simon: I'm not really sure that this is an example of (A). I think one could argue that that orientable surfaces with genera $g>1$ are the ones making up the generic infinite family, whereas $g=0$ and $g=1$ are the two "sporadic" examples. It is just that in this case, the sporadic examples came first. – José Figueroa-O'Farrill Mar 25 '11 at 1:18
How about classification of irreducible closed 2-manifolds: just $S^2, T^2, and RP^2$. All others are (connected) sums of these. So maybe this is an example of (B). @Jose why do you view g=1 as "sporadic"? (hyperbolic metric, presumably?) topologically they aren't much more special than their higher genus siblings. – Paul Mar 25 '11 at 2:48
@José - I think it depends on how you view the classification. Purely topologically, I would argue that the two families are: $A_g$ = the connected sum of $g$ copies of $T^2$, and $B_k$ = the connected sum of $k$ copies of $RP^2$. Ignoring the geometry, this is perfectly valid. The only dodgy part of it, perhaps, is that $B_0 = A_0$, but this sort of thing also occurs in the classification of Lie Algebras as well, so... – Simon Rose Mar 25 '11 at 14:51

There are a number of interesting classifications associated with tilings, e.g., the 17 wallpaper groups, the 230 space groups, or the 14 convex pentagons that tile the plane. (In the latter case I'm not sure if the classification has been rigorously proven to be complete.)

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String theories. For example, if you confine yourself to bosonic string theory you find it only works in dimension 26 despite the definition being completely independent of the number of dimensions. Similarly there are just 5 superstring theories. All of these theories are closely related to other interesting classifications in mathematics.

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There are four families for a connected Fatou Component $U$ (this is a component of the complement to the julia set obtained from some map $z \to P(z)/Q(z)$). The families are

• $U$ contains an attracting periodic point. (This is somehow very generic).
• $U$ contains a parabolic, (has an indifferent attracting point on the boundary).
• $U$ is a Siegel disk.
• $U$ is a Herman ring.

The two latter cases are somewhat special, and do not appear generically.

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As an example of (B), I'd mention Connes's classification of injective (type $II_{1}$) factors from the theory of von Neumann algebras. In this case, many apparently disparate constructions turn out to give a single object.

Classification of injective factors. Cases $II_1$, $II_\infty$, $III_\lambda$, $\lambda \not=1$. Ann. of Math. (2) 104 (1976), no. 1, 73-115.

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A1 Classification of linear spaces over a field. A2 Classification of indecomposable linear actions of Z (over any field) = Jordan blocks. B1 Classification of geometries (à la Thurston) in dimention 3 (?)

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Related to your B1, B2: Classification of two-dimensional geometries (Spherical, Euclidean and Hyperbolic). – Simon Rose Mar 24 '11 at 21:09

There are 5 platonic polyhedra...

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But these is a piece of a larger classification, which has a few exceptions and three families. – Mariano Suárez-Alvarez Mar 24 '11 at 19:34
Mariano: What's the third family? I can only think of the $A$ and $D$ families and the three platonic exceptions. – José Figueroa-O'Farrill Mar 25 '11 at 1:21
Simplices, n-cubes and generalized octahedra. – Jim Conant Mar 25 '11 at 1:53

Embedded CMC tori in the 3-sphere:

Andrews and Li (http://arxiv.org/pdf/1204.5007v3.pdf) have recently classified all embedded CMC (constant mean curvature) tori in the 3-sphere (up to isometries of $S^3$) by using methods from Brendles proof of the Lawson conjecture:

If one does not specify the (mean curvature) constant $H$, the space of embedded CMC tori is connected, but consist of infinitely many 1-dimensional families: The "exceptional" family consists of homogeneous CMC tori (parametrized by the mean curvature) which are (by definition) invariant under a 2-dimensional group of spherical isometries. At certain values of $H$ (namely $H=cot(\frac{\pi}{m})$) there bifurcates of another 1-dimensional family of CMC tori whose symmetry group is $S^1\times \mathbb Z_m.$

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In extensions of number fields $E/k$, you can look at primes in the ring of integers of $k$ that either (a) split in the integers of $E$, (b) remain prime in the integers of $E$, or (c) ramify. The case (c) consists of finitely many exceptions.

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The set of prime numbers is an example for (B): They are always presented as a single class, without any subfamilies.

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Twin prime conjecture? – Margaret Friedland Jun 16 '14 at 15:48
There are a lot of arguments in Diophantine equations using primes congruent to $1 \mod{6}$, primes congruent to $5 \mod{6}$, and two sporadic elements. Or you can have collections of families $\mod{n}$, with sporadic prime divisors of $n$. Maybe the most common: "Let $p$ be in the odd prime family..." – Zack Wolske Jun 16 '14 at 17:12