The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are:

(1) Classification of simply laced complex simple lie algebras.

(2) Finite subgroups of $Sl_2 (\mathbb{C})$

Are there any other objects that they classify?

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A nice quotation from P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob: Introduction to representation theory. arXiv:0901.0827: «If we needed to make contact with an alien civilization and show them how sophisticated our civilization is, perhaps showing them Dynkin diagrams would be the best choice!» – Mariano Suárez-Alvarez Nov 25 2009 at 12:49

Have you read this by John McKay and the 6 editions of This Week's Finds listed at the end? Week 230 gives plenty of ADE appearances.

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They classify quivers for which the path algebra is of finite representation type., acording to a famous theorem of P. Gabriel.

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They parametrize the finite dimensional preprojective algebras.

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They classify the (germs of) isolated rational singular points of two dimensional complex analytic spaces. See A. E. Durfee, Fifteen characterizations of rational double points and simple critical points. Enseign. Math. (2) 25 (1979), no. 1-2, 131--163.

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They classify the cluster algebras of finite type (that is, with a finite number of clusters). See S. Fomin, A. Zelevinski, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63--121.

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ADE correspond to skew-symmetric cluster algebras of finite type; you get the whole Dynkin (ABCDEFG) classification if you include all finite type cluster algebras (i.e., if you include ones that are skew-symmetrizable and not skew-symmetric). – Hugh Thomas Nov 25 2009 at 16:46

As Mariano said, the ADE Dynkin diagrams classify quivers of finite representation type. But wait, there's more. If you add one more vertex to a Dynkin diagram (in a particular way, not an arbitrary one), you get an extended Dynkin diagram (aka a Euclidian diagram). The extended ADE diagrams classify quivers of tame representation type. This is related to the fact that the extended ADE diagrams give you a positive semi-definite Tits form, while the ordinary ADE diagrams give you a positive definite Tits form.

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 I like the fact that, whereas subdiagrams of the Dynkin diagram classify Levis, subdiagrams of the Euclidean diagram classify semisimple centralisers (often called ‘pseudo-Levis’). – L Spice May 30 2011 at 2:24

They classify the so called "simple singularities" of differential maps, that is, of those types of singularities which involve no parameters. See V. I. Arnolʹd, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. 2, Chap. 15, sect. 1.

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They classify principal graphs of $II_1$-subfactors with index less than $4$. The principal graph can be $A_n$, $D_{2n}$, $E_6$, or $E_8$, but $D_{odd}$ and $E_7$ do not occur.

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And extended Dynkin diagrams partially classify the subfactors at index 4. – Scott Morrison Nov 4 2010 at 11:58

Let $G$ be a connected graph with the property that all eigenvalues of $G$ lie in $[-2,2]$ (such a $G$ is called cyclotomic). Then $G$ is either one of $\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$, an $\tilde{A}_n$ for $n\ge 2$, a $\tilde{D}_n$ for $n\ge4$, or an induced subgraph of one of these. In other words, the ADE graphs classify the maximal cyclotomic graphs.

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They classify certain types of rational conformal field theories, as in this recent review paper.

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This one is very directly related to the classification of subfactors with index less than 4! – Scott Morrison Nov 4 2010 at 11:58

This article gives a nice overview on the "ADE-problem". It is written in the late 1970's, so it does not cover more recent appearences. Where is their appearance is most surprising?

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Extended Dynkin diagrams appear naturally in Kodaira classification of singular fibers of an elliptic surface.

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You might also take a look at Slodowy, P. (1983), Platonic Solids, Kleinian Singularities and Lie Groups,Lecture Notes in Mathematics, No. 1008, pp. 102- 138,

Arnold's Trinities paper "Polymathematics : is mathematics a single science or a set of arts?", easily found on the net

and

Chapoton's own trinities page (also found on the net).

The last two focus on the "E" part of ADE, but give long lists of intriguing parallels.

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I suggest to take a look on a very nice Givental's paper (MR1138519 (92k:58031)):

"Reflection groups in singularity theory."

Here is the review (by V.D. Sedykh): The simple singularities of functions are classified by the Coxeter groups $(A,D,E$-classification). This classification arises in other problems, too (the classifications of the simple Lie algebras, of the finite quaternions groups and so on). The author gives a detailed survey of these results. He also considers the problems connected with the classification of the quasihomogeneous unimodular singularities of functions (the classification of the degenerations of elliptic curves, the theory of automorphic functions and so on) in this paper.

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