This Colloquium reviews a new class of exactly solvable models for systems exhibiting strong pairing correlations. There is an exact correspondence between these models and a two-dimensional, classical electrostatic problem, permitting a simple geometrical interpretation of the solutions. Applications are given to nuclei, metallic grains, and Bose-Einstein condensates.

Both equilibrium and nonequilibrium statistical-mechanical systems can display continuous phase transitions characterized by critical exponents. In equilibrium statistical mechanics, the factors leading to different values of the exponents, or universality classes, are known, but nonequilibrium systems are not nearly so well understood. This article summarizes the known universality classes for nonequilibrium statistical-mechanical lattice models.

In a number of materials such as the high-Tc cuprates, antiferromagnetic phases adjoin the superconducting phase. This led to the idea that the three components of the spin ordering and the two components of the superconducting order parameter can be unified into an effective Hamiltonian with an approximate symmetry of the five-dimensional rotation group, SO(5). As described in this article, support for this idea has been found in model systems as well as in experiment, particularly the observed coexistence of antiferromagnetism and superconductivity in magnetic fields.

The 2003 Nobel Prize in Physics was shared by A. A. Abrikosov, Vitaly L. Ginzburg, and Anthony J. Leggett. These lectures are the text of their addresses given on the occasion of the award.