We will give an informal talk about a single system of linear differential equations that are related to three distinct objects: non Calabi-Yau Hodge integrals, rubber integrals, and the quantum cohomology of the Hilbert scheme of points.

Forming moduli spaces of (polarised) algebraic varieties usually involves taking a quotient; one way to do this is by Geometric Invariant Theory. Except in some special cases the varieties that this parametrises, i.e. the (semi)stable points of the group action, have not been identified. This talk will describe joint work with Julius Ross attempting to describe these stable algebraic varieties; the stability criterion turns out to be similar to the more familiar one for bundles (which will be reviewed). Another motivation for this subject is the Hitchin-Kobayashi correspondence. Bundle stability is related to the existence of Hermitian-Yang-Mills connections; similarly stability of varieties is conjecturally related to the existence of Kahler-Einstein and constant scalar curvature Kahler metrics.