| Summary: | The main subject of this Thesis is the study of harmonic maps from compact Riemann surfaces into unitary groups, and various generalisations and related subjects. Harmonic maps are critical points of the energy functional. In the case we are considering, the associated Euler-Lagrange equations are particularly simple, because of the conformal invariance of the energy for maps from surfaces, which emphasizes the role of the complex structure, and of the simplicity of the target manifold. Another important feature is that the non-linear equations are representable as O-curvature conditions for families (loops) of connections. This is the elliptic version of a phenomenon which is typical of a class of evolution equations, where it induces soliton behaviour, and complete integrability. ln this elliptic situation, this representation (due to Zakharov et al.) allows us to substitute algebraic geometry to analysis, in the description of the solutions.
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