Alexander O. Ivanov

Lecture 1

Hausdorff and Gromov–Hausdorff distances

1. Definition and basic properties of the Hausdorff distance
2. The Hausdorff distance is a generalized metric on closed subsets
3. Gromov–Hausdorff distance between metric spaces
   1. GH-distance in terms of embeddings
   2. GH-distance in terms of correspondences
4. Elementary properties of the GH-distance

Lecture 2

Properties of the Gromov–Hausdorff distance

1. GH-distance is a generalized pseudometric
2. Completeness and intrinsicness of the GH-distance
3. Technique of irreducible correspondences
4. Examples
   1. 3-point spaces
   2. Spaces with one non-zero distance (simplexes)
5. Continuity of metric characteristics and closeness of subclasses
   1. Continuity of functionals: diameter, density, covering numbers
   2. Semi-continuity of density and covering numbers
   3. Closedness of classes: separable, totally bounded and compact spaces
   4. Relation to uniform convergence of metrics
   5. Examples of calculations

Lecture 3

Compact metric spaces in GH-topology

1. Metric on ℳ
2. Completeness of ℳ
3. Separability of ℳ
4. Gromov's criterion for precompactness
5. Geodesicity of ℳ
6. Gromov–Hausdorff distance to simplexes
   1. Definition of a simplex
   2. Distance to a simplex with a larger number of points
   3. Distance to a simplex with less or equal number of points
   4. Distance to a simplex with an infinite number of points
   5. Example: the distance to a simplex for a two-point space

Lecture 4

Local and global isometries of space ℳ

1. Problem statement
2. Local structure of ℳ in a neighborhood of generic spaces
3. Invariance of subsets
4. Local affinity of isometries
5. Global triviality of isometries
6. Consequences

Lectures 1 and 2 — files

The first two lectures of the minicourse: the lecture text, the seminar problems and the problem sheets.

Lectures 1 and 2 — lecture text (PDF)

Lectures 1 and 2 — seminar problems (PDF)

Lectures 1 and 2 — problem sheets (PDF)

Lectures 2–4 — files

The remaining lectures of the minicourse: the lecture text for lectures 2–4, and the seminar problems and problem sheets for lectures 3–4.

Lectures 2–4 — lecture text (PDF)

Lectures 3–4 — seminar problems (PDF)

Lectures 3–4 — problem sheets (PDF)