  The article describes a topological theory of quasiperiodic functions on the
plane. The development of this theory was started (in different terminology) by
the Moscow topology group in early 1980s. It was motivated by the needs of
solid state physics, as a partial (nongeneric) case of Hamiltonian foliations
of Fermi surfaces with multivalued Hamiltonian function. The unexpected
discoveries of their topological properties that were made in 1980s and 1990s
have finally led to nontrivial physical conclusions along the lines of the
so-called geometric strong magnetic field limit. A very fruitful new point of
view comes from the reformulation of that problem in terms of quasiperiodic
functions and an extension to higher dimensions made in 1999. One may say that,
for single crystal normal metals put in a magnetic field, the semiclassical
trajectories of electrons in the space of quasimomenta are exactly the level
lines of the quasiperiodic function with three quasiperiods that is the
dispersion relation restricted to a plane orthogonal to the magnetic field.
General studies of the topological properties of levels of quasiperiodic
functions on the plane with any number of quasiperiods were started in 1999
when certain ideas were formulated for the case of four quasiperiods. The last
section of this work contains a complete proof of these results. Some new
physical applications of the general problem were found recently.
