Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one correspondence with the collection of simply laced Dynkin diagrams. The simply laced Dynkin diagrams have themselves been classified into two infinite families (denoted $A_n$ and $D_n$) and three exceptional examples (denoted E_6, E_7, and E_8). See the wikipedia page for more details.

There is a veritable laundry list of objects that admit an ADE classification. Examples include (modulo a number of qualifiers that I don't want to get into):

• Semisimple Lie algebras
• Conformal field theories
• Tame quivers
• Platonic solids
• Positive definite quadratic forms on graphs

The list goes on. Accoding to the aforementioned wikipedia page, Vladmir Arnold asked in 1976 if there is a connection between these different kinds of objects which really explains why they all admit a common classification. The page also makes an offhand comment about how such a connection might be suggested by string theory.

I am hoping that somebody can explain some of the progress that has been made (if any) on Arnold's question. A good answer to this question is not one which explains the proofs that various different types of objects admit ADE classifications, nor one which aimlessly extends the list above. Rather, I would like to see someone take a collection of objects which on the surface are unrelated but which all have ADE classifications and then outline a deeper connection between them which at least suggests that they might all have a common classification. Bonus points if anyone can justify wikipedia's invocation of string theory in this context.

Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one correspondence with the collection of simply laced Dynkin diagrams. The simply laced Dynkin diagrams have themselves been classified into two infinite families (denoted $\mathrm{A}_n$ and $\mathrm{D}_n$) and three exceptional examples (denoted $\mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$). See the Wikipedia page for more details.

There is a veritable laundry list of objects that admit an ADE classification. Examples include (modulo a number of qualifiers that I don't want to get into):

• Semisimple Lie algebras
• Conformal field theories
• Tame quivers
• Platonic solids
• Positive definite quadratic forms on graphs

The list goes on. Accoding to the aforementioned wikipedia page, Vladmir Arnold asked in 1976 if there is a connection between these different kinds of objects which really explains why they all admit a common classification. The page also makes an offhand comment about how such a connection might be suggested by string theory.

I am hoping that somebody can explain some of the progress that has been made (if any) on Arnold's question. A good answer to this question is not one which explains the proofs that various different types of objects admit ADE classifications, nor one which aimlessly extends the list above. Rather, I would like to see someone take a collection of objects which on the surface are unrelated but which all have ADE classifications and then outline a deeper connection between them which at least suggests that they might all have a common classification. Bonus points if anyone can justify wikipedia's invocation of string theory in this context.