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-Carathéodory 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.

Inhalt

• Introduction to the Notes of the School on Analysis and Geometry in Metric Spaces. - Geometric Inequalities on Riemannian and Sub-Riemannian Manifolds by Heat Semigroups Techniques. - Differentiation of Measures in Metric Spaces. - Sobolev Spaces in Extended Metric-Measure Spaces. - Brief Survey on Functions of Bounded Variation (BV) in Metric Setting.