Faces of matrix models

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as $\tau$-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of $W$-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic “special functions” to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.