Banach algebras in inverse problems of mathematical physics

Inverse problems of mathematical physics deal with reconstructing the structure and/or parameters of a system from its reaction to external forces (seismology, acoustic and impedance tomography, etc.) Some time ago (in 2003) we discovered a deep and unexpected relation of such probelms with representing an abstract commutative Banach algebra as an algebra of continuous functions on a compact space (geometrization of rings in Gelfand's sense). It has turned out that this representation (the Gelfand transform) indeed solves some important inverse problems. Soon afterwards, the list of such probelms and algebras helping to solve them was extended. In this talk, we illustrate such relations by examples, discuss some new results and open problems.