Alexander Its

Operator Valued Riemann-Hilbert Problems. Then and Now.

In the context of integrable systems, operator valued Riemann-Hilbert problems first appeared in the late 80s early 90s work of A. Izergin, V. Korepin, N. Slavnov and the speaker on the asymptotic analysis of quantum correlation functions. It turns out that the transition to the operator valued Riemann-Hilbert setting manifests the transition from free fermion to non free fermion exactly solvable quantum models. Very recently, the operator valued Riemann-Hilbert problems started to show up in the problems related to the integrable probability; notably, in the description of the important solutions of the KPZ equations. These are the works of I. Corwin, P. Ghosal, T. Bothner,
M. Cafasso, and S. Tarricone.

The principal goal of the talk is to remind the old, still unsolved important questions associated with the quantum operator valued Riemann-Hilbert problems and to highlight the new challenges emerged in the area due to its involvement in the probabilistic applications. (All the basic aspects related to the theory of quantum correlators will be explained.)