From classical to quantum integrability, and back

Hirota relations in their various incarnations play an important role in both classical and quantum integrable systems, from matrix integrals and PDE's to one-dimensional quantum spin chains and two dimensional quantum field theories (QFT). The Wronskian solutions of discrete Hirota equations (T-systems) are related to the symmetry of these systems. They can be used, when supplied with analyticity properties, to find exact energy spectra of quantum spin chains and QFT's in finite volume. The method will be demonstrated on recently discovered Riemann-Hilbert spectral equations for the most emblematic example of AdS/CFT correspondence - the 4-dimensional N=4 Super-Yang-Mills theory.