Finite, Tame, Wild

Denial

There is a trichotomy in representation theory due to Drozd which roughly says that the problem of understanding the (finite dimensional) representations of a finite-dimensional algebra is either finite, tame, or wild.

• Finite type algebras have only finitely many indecomposable representations up to isomorphism.
• Tame type algebras have infinitely many indecomposables but they can be classified in a reasonable way. For a fixed dimension d, all but finitely many indecomposable representations with dimension less than d are grouped into a finite number of one-parameter families.
• Wild type algebras have infinitely many indecomposable representations up to iso and the problem of classifying them is ‘hopeless’.

The use of the word ‘hopeless’ stuck with me when I first saw it used to describe these wild problems. Initially it was because I didn’t really know what they meant. Hopeless how? Are people just not trying hard enough? I did not believe that these problems were truly hopeless like claimed.

Anger

Bargaining

Depression

Acceptance

Reflection

I’ve used the analogy with the 5 stages of grief partly because it’s a little amusing and partly because there is some truth to it. That then begs the question: What is it that I’m grieving about? To grieve is to “feel intense sorrow” and sorrow is “a feeling of deep distress caused by loss, disappointment, or other misfortune”. What misfortune have I suffered? What, if anything, have I lost?

One answer, perhaps, is that I have lost the belief that with enough time and effort all of these problems could be understood completely. The only justifications I had for believing such a thing

Dramatic line about the fact that my existential dread concerning wild problems was only fully realised after actually working with a couple tame problems for a while: Only when you peek over the edge does the depth of the abyss become clear.