Lectures will be devoted to harmonic differential forms, a classical
subject playing an important role in Analysis, Geometry and Mathematical
Physics. Harmonic differential forms will be considered as
multidimensional analogs of analytic functions of one complex variable.
This point of view leads to interesting problems suggested by Complex
Analysis and related to harmonic forms. Approximation properties of
harmonic forms will be discussed (parallel to the theory of rational
approximation in the complex plane). Another theme is uniqueness
properties of harmonic forms. These themes are traditional and
practically exhausted for harmonic forms of degree one in the plane
(i.e. for complex analytic functions). But in higher dimensions (even in
dimension three, i.e. for curl-free and divergence-free vector fields)
they generate a lot of natural and hard questions (most of them open)
and require new tools and ideas.