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A quiver is a basic combinatorial object, consisting of vertices and arrows between them. A quiver defines in an obvious way a category.
So, given a quiver Q and an abelian category A, one may investigate representations of Q in the category A, i.e., functors from the category assigned to Q to A.
The best studied case is when A is the category of finite dimensional vector spaces over a field k.
Classification problems for quiver representations include the problem of the Jordan normal form as well as the
representation theory of finite dimensional kalgebras.
If A is the category of coherent sheaves over, say, a complex projective manifold X,
one arrives at the notion of a quiver sheaf. (In fact, there is a twist to this, in the literal sense of the word.)
Famous examples of (twisted) quiver sheaves are Higgs bundles. They are associated with the quiver Q consisting of one
vertex and one arrow joining that vertex to itself.
In the talk, we will briefly review the language of quivers, representations of quivers in the category of finite dimensional
vector spaces, and their moduli spaces. Then, we will survey various results on (twisted) quiver sheaves.
We plan to address (some of) the following topics: -- semistability and moduli spaces, -- geometry of moduli spaces, -- topology of moduli spaces of Higgs bundles. |