Abstract:
In the first part of the lectures, we will work through the definition and basic geometric properties of moduli spaces of representations of quivers, thereby recalling necessary facts from the representation theory of quivers and from Geometric Invariant Theory.
In the second part, we will focus on their cohomological properties, deriving formulas for (generating functions of) Poincare polynomials and Euler characteristics.
In the third part (depending on available time and participants' interests) relations between quiver moduli, noncommutative algebraic geometry and motivic Donaldson-Thomas theory will be discussed.
Suggested reading:
1) For basics of the representation theory of quivers, see:
W. Crawley-Boevey: Lectures on representations of quiver. http://www1.maths.leeds.ac.uk/~pmtwc/quivlecs.pdf
2) The necessary methods from Geometric Invariant Theory are covered in:
S. Mukai, An Introduction to Invariants and Moduli. Cambridge Studies in Advanced Mathematics 81, Cambridge University Press, 2003.
3) The construction and basic geometric facts on quiver moduli can be found in:
A. King, Moduli of representations of finite-dimensional algebras. Quart. J. Math.Oxford Ser. (2) 45 (1994), 180, 515–530.
4) A survey paper (written for representation theorists) covering most of the material of the lectures:
M.R.: Moduli of representations of quivers.
http://de.arxiv.org/abs/0802.2147
For the potential topics of the third part:
5) The relation to Donaldson-Thomas theory is worked out in:
M.R.: Poisson automorphisms and quiver moduli. http://de.arxiv.org/abs/0804.3214
6) Motivic Donaldson-Thomas theory itself is developed in:
M. Kontsevich, Y. Soibelman: Motivic Donaldson-Thomas invariants: summary of results
http://de.arxiv.org/abs/0910.4315
7) Quivers and quiver moduli play a role in the noncommutative algebraic geometry developed in:
L. Le Bruyn: Noncommutative Geometry and Cayley-smooth Orders, Pure and Applied Mathematics 290, Chapman and Hall / CRC, 2007. http://win.ua.ac.be/~lebruyn/LeBruyn2007a.pdf