Geometry of Metric Spaces
Metric spaces are one of most elementary notions of geometry. Gromov discovered that many advanced notions of differential geometry can be interpreted in terms of the metric.
I will explain the CAT conditions (positive and negative curvature) and their consequences, such as the Cartan-Hadamard theorem: every path metric space with non-positive curvature is contractible. If time permits, I would cover the theory of Gromov hyperbolic spaces and its applications to group theory (the notion of the hyperbolic group).
The course should be accessible to anybody with some understanding of point-set topology (topological space, continuous maps, compactness).
1. Metric spaces, path metric, geodesics, Hopf-Rinow theorem.
2. CAT-inequalities, CAT(0)-spaces, Cartan-Hadamard theorem.
3. Gromov hyperbolic spaces, hyperbolic groups, quasi-isometry of metric spaces. Main examples of hyperbolic and non-hyperbolic groups.
4. Isoperimetric inequality and algorithmic solvability of word problem in hyperbolic groups.