A general framework for dualities

The aim of this talk is to provide a general setting in which a number of important dualities in mathematics can be framed uniformly.  The setting comes about as a natural generalisation of the Galois connection between ideals of polynomials with coefficients in a field K and affine varieties in K^n.  The general picture that comes into sight is that the topological representations of Stone, Priestley, Baker-Beynon, Gel’fand, or Pontryagin are to their respective classes of structures just as affine varieties are to K-algebras.