This book is devoted to classical and modern achievements in complex analysis. In order to
benefit most from it a first-year university background is sufficient all other statements
and proofs are provided.We begin with a brief but fairly complete course on the theory of
holomorphic meromorphic and harmonic functions. We then present a uniformization theory and
discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as
a factor space of a contracted space by a discrete group. Next we consider compact Riemann
surfaces and prove the classical theorems of Riemann-Roch Abel Weierstrass etc. We also
construct theta functions that are very important for a range of applications.After that we
turn to modern applications of this theory. First we build the (important for mathematics and
mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at
important solutions to these differential equations. We subsequently use the theory of harmonic
functions and the theory of differential hierarchies to explicitly construct a conformal
mapping that translates an arbitrary contractible domain into a standard disk - a classical
problem that has important applications in hydrodynamics gas dynamics etc.The book is based
on numerous lecture courses given by the author at the Independent University of Moscow and at
the Mathematics Department of the Higher School of Economics.
