Concise guide for electronic topological transitions

In this short review we pass through the milestones in the studies of the electronic topological transitions (ETT) and focus on some recent applications of the ideas worked out in their classical theory. These are: two-dimensional electron systems, de Haas-van Alphen effect, classification of ETT in multidimensional systems, superconductivity in systems close to ETT, and thermoelectricity in heavy-fermion systems where the cascades of topological changes of Fermi surface (FS) are generated by magnetic field. The history of studies of ETT is inextricably linked with Kharkov school of condensed matter physics, with such names as I. M. Lifshitz, V. G. Bar'yakhtar, and many other. Among them is Moisey Isaakovich Kaganov, who contributed much in the studies of the role of geometry and topology of FS in physical properties of the metals. Two of the authors (A.V. and Y.G.) had the honor and pleasure to work with M. I. Kaganov - Y. M. Blanter, M. I. Kaganov, A. V. Pantsulaya, and A. A. Varlamov, Phys. Rep. 245, 159 (1994);(1) all of us have been learning the niceties of science from his books. "The Fermi surface is the stage on which the drama of the life of the electron is played out" wrote Kaganov and Lifshitz. We devote this work to their memory.