Geometry III: Discrete conformal maps (Winter 2015)

This is a course of the Berlin Mathematical School held in English.

Contents

The course will be on the following two topics and what they have to do with each other: • Nonlinear theory of discrete conformal maps: circle packings and circle patterns, discrete conformal equivalence of triangle meshes. • Realization of hyperpolic polyhedra with prescribed dihedral angles or intrinsic metric.

Literature

• W. P. Thurston. The Geometry and Topology of Three-Manifolds, electronic edition of the 1980 lecture notes distributed by Princeton University.
• J. Milnor. Hyperbolic geometry: The first 150 years. Bull. Amer. Math. Soc. (N.S.) 6, (1982), no. 1, 9-24.
• I. Rivin. A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. (2), 143 (1996), no. 1, 51-70.
• Y. Colin de Verdière. Un principe variationnel pour les empilements de cercles. Invent. Math. 104 (1991), no. 1, 655-669.
• I. Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2), 139 (1994), no. 1, 553-580.
• I. Rivin. Combinatorial optimization in geometry. Adv. in Appl. Math. 31 (2003), no. 1, 242-271. arXiv
• A. I. Bobenko, B. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689.
  
• B. Springborn. Variational principles for circle patterns. Doktorarbeit, TU Berlin, 2003.
• F. Guéritaud. On an elementary proof of Rivin's characterization of convex ideal hyperbolic polyhedra by their dihedral angles. Geometriae Dedicata 108 (2004), no. 1, 111-124.
• H. Kneser. Der Simplexinhalt in der nichteuklidischen Geometrie. Deutsche Mathematik 1 (1936), 337-334.
• J. Böhm, E. Hertel. Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung. Mathematische Monographien, Bd. 14. VEB Deutscher Verlag der Wissenschaften, Berlin, 1980. Lizenzausgabe bei Birkhäuser 1981.
• D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov. Geometry of spaces of constant curvature. In: E. B. Vinberg (editor). Geometry II. Encyclopedia of Mathematical Sciences 29. Springer, Berlin, 1993. Pages 1-138.
• J. Milnor. The Schläfli differential equalitiy. In: Collected Papers, volume 1, pages 281-295. Publish or Perish Inc., Houston, TX, 1994.
• A. Bobenko, U. Pinkall, B. Springborn. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19-4 (2015) 2155-2215. arXiv:1005.2698.
• L. Lewin. Polylogarithms and associated functions. North-Holland Publishing Co., New York-Amsterdam, 1981.

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