Metrische Geometrie

Vorbesprechung und Themenvergabe am Donnerstag, 21. Februar, 15:15 Uhr im HG E 33.1.

Die Kurssprache ist Deutsch, die Literatur allerdings ausschliesslich auf Englisch.

Themenliste

1. Length spaces: basic definitions, geodesics and the Hopf-Rinow theorem. Source: [1, Section 1 & 2]
2. Constructions of metric spaces. Source: [1, Section 3]
3. Groups actions and coverings. Source: [1, Section 4 & 5]
4. Gromov-Hausdorff distance and Gromov compactness criterion. Source: [2, Section 1 & 2]
5. Spaces of bounded curvature and Toponogov's comparison theorem. Source: [4, Chapter 4], [3]
6. Spaces of curvature bounded above and Reshetniak's glueing theorem. Source: [4, Chapter 9]
7. CAT(0)-Spaces and the Globalization theorem. Source: [4, Chapter 9]
8. Flat strips. Source: [5, Chapter II.2]
9. Billiards. Source: [7], [4, Chapter 9]
10. Flat planes in spaces of nonpositive curvature. Source: [6]
11. CAT(0) cube complexes and tile sets. Source: [8]

Literatur

• [1] U. Lang, Length Spaces, Lecture Notes, 2004
• [2] U. Lang, Notes on Rectifiability, Lecture Notes, 2006
• [3] U. Lang and V. Schroeder, On Topogonov's comparision theorem for Alexandrov spaces, ArXiv e-prints, 2012
• [4] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001
• [5] M. Bridson and A. Heafliger, Metric Spaces of Non-Positive Curvature, Springer 1999
• [6] M, Bridson, On the existence of flat planes in spaces of nonpositive curvature, Proc. Amer. Math. Soc. 123 (1995), p. 223-235
• [7] D. Burago, S. Ferleger and A. Kononenko, A geometric approach to semi­dispersing billiards (Survey). Ergodic Theory and Dynamical Systems 18 (1998), p. 303­-319
• [8] J. Kari and P. Papasoglu, Deterministic aperiodic tile sets, Geom. funct. anal. 9 (1999), p. 353-369