Toda hierarchies and their applications

The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as reductions. These integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s. A recent example is a series of studies on models of statistical mechanics called the melting crystal model. This research has revealed that the aforementioned two reductions of the 2D Toda hierarchy underlie two different melting crystal models. Technical clues are a fermionic realization of the quantum torus algebra, special algebraic relations therein called shift symmetries, and a matrix factorization problem. The two melting crystal models thus exhibit remarkable similarity with the Hermitian and unitary matrix models for which the two reductions of the 2D Toda hierarchy play the role of fundamental integrable structures.