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The wave maps system is one of the most beautiful and challenging
nonlinear hyperbolic systems, which has captured the attention of
mathematicians for more than thirty years now. In the study of its
various issues, such as the well-posedness theory, the formation of
singularities, and the stability of the solitons, in order to
obtain optimal results, one has to use intricate tools coming not
only from analysis, but also from geometry and topology. Moreover,
the wave maps system is nothing other than the Euler-Lagrange
system for the nonlinear sigma model, which is one of the
fundamental problems in classical field theory. One of the goals of
our book is to give an up-to-date and almost self-contained
overview of the main regularity results proved for wave maps.
Another one is to introduce, to a wide mathematical audience,
physically motivated generalizations of the wave maps system (e.g.,
the Skyrme model), which are extremely interesting and difficult in
their own right.
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