the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact. Hirota bilinear equations and τ-functions Another viewpoint that arose in the modern theory of integrable systems originated in a calculational approach pioneered by Ryogo Hirota, which involved replacing the original nonlinear dynamical system with a bilinear system of constant coefficient equations for an auxiliary quantity, which later came to be known as the -function. These are now referred to as Hirota equations. Although originally appearing just as a calculational device, without any clear relation to the inverse scattering approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as solitons could be derived. Subsequently, this was beautifully interpreted, by Mikio Sato and his students, at first for the case of integrable hierarchies of PDE's, such as the Kadomtsev-Petviashvili hierarchy, but then for much more general classes of integrable hierarchies, as a sort of universal phase space approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian group action on a (finite or infinite) Grassmann manifold. The -function was viewed as the determinant of a projection operator from elements of the group orbit to some origin within the Grassmannian, and the Hirota equations as expressing the Plucker relations, characterizing the Plücker embedding of the Grassmannian in the projectivizatin of a suitably defined (infinite) exterior space, viewed as a fermionic Fock space. Quantum integrable systems There is also a notion of quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to local conservation laws. Every Hamiltonian has an infinite set of conserved quantities given