Geometry III: Discrete conformal maps (Winter 2019)

This is a course of the Berlin Mathematical School held in English (unless all attending BMS students prefer German).

Contents

The course will be on the following 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.

Movies

• conform. A short film about conformal maps. [youtube, vimeo]
• conform. Ein kurzer Film über konforme Abbildungen. (deutsch) [vimeo]
• Koebe polyhedra and minimal surfaces. [youtube]

Literature

• 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.
• A. Bobenko, U. Pinkall, B. Springborn. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19-4 (2015) 2155-2215. arXiv:1005.2698.
• A. I. Bobenko, B. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689.
  
• 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.
• Y. Colin de Verdière. Un principe variationnel pour les empilements de cercles. Invent. Math. 104 (1991), no. 1, 655-669.
• 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.
• L. Lewin. Polylogarithms and associated functions. North-Holland Publishing Co., New York-Amsterdam, 1981.
• J. Milnor. Hyperbolic geometry: The first 150 years. Bull. Amer. Math. Soc. (N.S.) 6, (1982), no. 1, 9-24.
• J. Milnor. The Schläfli differential equalitiy. In: Collected Papers, volume 1, pages 281-295. Publish or Perish Inc., Houston, TX, 1994.
• M. Passare. How to compute Σ1/n² by solving integrals. Amer. Math. Monthly 115 (2008), no. 8, 745-752.
• I. Rivin. A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. (2), 143 (1996), no. 1, 51-70.
• 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
• B. Springborn. A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z., 249(3):513--517, 2005.
• B. Springborn. Variational principles for circle patterns. Doktorarbeit, TU Berlin, 2003.
• B. Springborn. Ideal Hyperbolic Polyhedra and Discrete Uniformization. Discrete Comput. Geom. (2019)
• W. P. Thurston. The Geometry and Topology of Three-Manifolds, electronic edition of the 1980 lecture notes distributed by Princeton University.
• D. Zagier. The dilogarithm function. In: Frontiers in number theory, physics, and geometry. II, 3-65, Springer, Berlin, 2007.

Office hours