Algebras and functional models of operators and systems in inverse problems of Mathematical Physics

The talk presents the main ideas and recent results one of an approaches to inverse problems – the Boundary Control Method (BCM). The BCM is a pronounced interdisciplinary approach that uses deep (sometimes quite unexpected) connections between various branches of mathematics. In their including: control theory and systems theory, asymptotic methods (geometric optics), Riemannian geometry, Banach algebras and $C^*$-algebras, operator theory (functional models), etc. Among the results there are the solution of the 2-dimensional impedance tomography problem tomography for non-orientable surfaces, wave model of metric spaces and operators, de Branges spaces in light of the “time-frequency” dualism, etc.