Part 1 Historical survey and introduction to the theory of minimal surfaces: the origins of multidimensional variational calculus; the nineteenth century, the era of the discovery of basic minimal surface properties; topological and physical properties of minimal surfaces; the Plateau principles - minimal surfaces in animate nature. Part 2 Survey of some important publications in minimal surface theory from the nineteenth to the early twentieth century: Monge (1746-1818); Poisson (1781-1840); Plateau (1801-1883); some works of the early twentieth century Rado, Douglan; minimal surfaces in the large. Part 3 Some facts from elementary topology: singular and cellular homology groups; cohomology groups and obstructions to the extensions of mappings. Part 4 Modern state of minimal surface theory: minimal surfaces and homology; integral currents; minimal currents in Riemannian manifolds; minimization of volumes of manifolds with fixed boundary and of closed manifolds - existence of a minimum in each spectral bordism class; generalized homology and cohomology theories and their relation to the multidimensional Plateau problem; existence of a minimum in each homotopy class of multivarifolds; cases where a solution of the Dirichlet problem for the equation of minimal surfaces of high codimensions does not exist; example of a smooth, closed, unknotted, curve in R3 bounding only minimal surfaces of large genus; certain new methods of effective construction of globally minimal surfaces in Riemannian manifolds; totally geodesic surfaces realizing non-trivial cycles, cocycles, and elements of homotopy groups in symmetric spaces; Bott periodicity and its relation with the multidimensional Dirichlet functional; survey of some recent results in harmonic mapping theory.
