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).

Program.

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.

The course should be accessible to anybody with
some understanding of point-set topology (topological
space, continuous maps, compactness).

Literature:
Burago, D., Burago, Y., and Ivanov, S. A course in metric geometry,
AMS Graduate Studies in Mathematics Volume 33 (2001).

Gromov, M., Metric structures for Riemannian and
non-Riemannian spaces. Progress in Mathematics. Vol. 152.
With appendices by M. Katz, P. Pansu, and
S. Semmes. Boston,  Birkhauser, 1999

Gromov, M., Sign and geometric meaning of curvature,
Rend. Semin. Mat. Fis. Milano 61 (1991), 9-123
https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/177.pdf

http://homepages.math.uic.edu/~mbhull/hyperbolic%20lecture%20notes.pdf

https://www.math.ucdavis.edu/~kapovich/280-2020/vaisala.pdf

Martin R. Bridson, Andre Hafliger,
Metric Spaces of Non-Positive Curvature,
2011, Springer