Abstract: I’ll discuss some open questions concerning representations of the Kronecker quiver. By now it’s fair to say we understand the finite dimensional representations fairly well, so we’ll concentrate on what can be said about understanding the representations of infinite dimension. There are a number of challenges still remaining and, as it happens, we know a lot less about the Kronecker quiver (in some senses) than we do about polynomial rings for instance. I’ll make precise what it might mean to ‘understand’ the infinite dimensional representations, give an overview of what I do know, discuss what I hope is tractable, and try to indicate why better understanding the Kronecker quiver is a good warmup for a number of important problems in representation theory and geometry.
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