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This book includes four courses on geometric measure theory, the
calculus of variations, partial differential equations, and
differential geometry. Authored by leading experts in their fields,
the lectures present different approaches to research topics with
the common background of a relevant underlying, usually
non-Riemannian, geometric structure. In particular, the topics
covered concern differentiation and functions of bounded variation
in metric spaces, Sobolev spaces, and differential geometry in the
so-called Carnot-Caratheodory spaces. The text is based on lectures
presented at the 10th School on "Analysis and Geometry in Metric
Spaces" held in Levico Terme (TN), Italy, in collaboration with the
University of Trento, Fondazione Bruno Kessler and CIME, Italy. The
book is addressed to both graduate students and researchers.
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