The collection contains two series of lectures delivered at the X Summer School of Mathematics . Topics of the first cycle of lectures /g. Nalchik, July 2-16, 1972 . / - some questions of functional analysis and mathematical physics. Lectures are devoted to piecewise-polynomial approximation of summable functions; their applications to the functions of Sobolev classes and a number of problems of analysis and spectral theory; some general approximate methods of solving linear, and nonlinear operator equations. Lectures of the second cycle /pos. Katsiveli, 25 June 15 July 1972 . / are two large courses: formal groups and the functions of many complex variables. The lectures reflected some new results on the use of formal groups. A new approach to the topology of smooth manifolds based on the theory of formal groups Lectures are designed for undergraduate students, as well as non-specialists interested in modern mathematical theories.