The representations of the fundamental group of an algebraic variety form important topological invariants connecting group theory with geometry. In this talk, we focus on the varieties of smallest non-trivial dimension: curves. Using quivers and jet schemes, we show that the representation variety has mild singularities. We apply this to show that the number of irreducible complex representations of SL_n(Z) of dimension at most m grows at most as the square of m, for a fixed n>2.