  This series of lectures is planned as a generalization of author's large
(more than fifteen years) experience of work in the theoretical physics. The
modern theoretical physics is based on the group-theoretical approach which
generates the formalism of the principal fibre bundles and the instanton
approach. The latter is based on the Pontrjagin's degree of map theorem and
this theorem is the original ``bridge'' between homology and cohomology
theories. The author plans to devote his two first lectures to fibre bundle
theory: this is the foundation on which the modern physics rests -- the theory
of gauge groups and the Yang-Mills fields. The idea of connection and curvature
(first of all for the principal fibre bundles) will be given also. The lectures
are devoted to the Pontrjagin's degree of map theorem, to the theories of
monopoles and instantons, to the theory of the topological index of the
elliptical operator. The information accumulated to this moment allows us to
apply these theories to some questions of conformal anomalies and to the
topological aspects of QCD. More comprehensive questions of modern topology
(for example the algebraic (co)homology theories, the theory of the spectral
sequences) will be expounded in the further lectures.
