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Geometry III: Discrete conformal maps
(Winter 2022/23)
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[Bobenko/Sechelmann/Springborn]
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[Gillespie/Springborn/Crane]
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Lectures
Boris Springborn
Mon |
12-14 |
MA 851 |
Fri |
10-12 |
MA 751 |
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 their connections:
- 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 prescribed intrinsic metric.
Shared students' notes for participants (sorry, password protected)
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
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Circle packings, circle patterns,
ideal hyperbolic polyhedra with prescribed dihedral angles,
discrete conformal maps via circumcircle intersection angles
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A. I. Bobenko, B. Springborn. Variational principles for circle
patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689.
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Y. Colin de Verdière. Un principe variationnel pour les empilements de
cercles. Invent. Math. 104 (1991), no. 1,
655-669.
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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.
- I. Rivin. A characterization of ideal polyhedra
in hyperbolic 3-space, Ann. of Math. (2), 143 (1996), no. 1, 51-70.
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I. Rivin. Euclidean structures on simplicial surfaces
and hyperbolic volume. Ann. of Math. (2), 139 (1994), no. 1, 553-580.
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I. Rivin. Combinatorial optimization in geometry. Adv. in Appl. Math. 31 (2003), no. 1,
242-271. arXiv
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B. Springborn. A unique representation of polyhedral types. Centering via
Möbius transformations. Math. Z., 249(3):513--517, 2005.
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B. Springborn. Variational principles for circle patterns.
Doktorarbeit, TU Berlin, 2003.
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W. P. Thurston. The Geometry and Topology of
Three-Manifolds, electronic edition of the 1980 lecture notes distributed by Princeton University.
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Discrete conformal maps via length scaling, ideal hyperbolic polyhedra with prescribed intrinsic metric
- A. Bobenko, U. Pinkall, B. Springborn. Discrete conformal
maps and ideal hyperbolic polyhedra. Geom. Topol. 19-4
(2015) 2155-2215. arXiv:1005.2698.
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A. I. Bobenko, S. Sechelmann, and B. Springborn. Discrete conformal maps:
boundary value problems, circle domains, Fuchsian and Schottky uniformization. In A. I. Bobenko, editor,
Advances in Discrete Differential Geometry, pages 1--56. Springer, Berlin, 2016.
- F. Luo. Combinatorial Yamabe flow on surfaces.Commun. Contemp. Math. 6 (2004), no. 5, 765–780.
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M. Gillespie, B. Springborn, and K. Crane. Discrete conformal equivalence
of polyhedral surfaces. ACM Trans. Graph., 40(4), 2021.
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X. D. Gu, F. Luo, J. Sun, T. Wu. A discrete uniformization theorem for polyhedral surfaces.
J. Differential Geom. 109 (2018), no. 2, 223–256.
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B. Springborn. Ideal Hyperbolic Polyhedra and Discrete Uniformization.
Discrete Comput. Geom. (2019)
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General / related / background
- 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.
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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.
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H. Kneser. Der
Simplexinhalt in der nichteuklidischen Geometrie. Deutsche Mathematik 1 (1936), 337-334. (good paper in nasty journal)
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L. Lewin. Polylogarithms and associated functions.
North-Holland Publishing Co., New York-Amsterdam, 1981.
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J. Milnor. Hyperbolic geometry: The first 150
years. Bull. Amer. Math. Soc. (N.S.) 6,
(1982), no. 1, 9-24.
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J. Milnor. The Schläfli differential equalitiy. In: Collected
Papers, volume 1, pages 281-295. Publish or Perish Inc., Houston, TX, 1994.
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M. Passare. How to compute Σ1/n2 by solving integrals.
Amer. Math. Monthly 115 (2008), no. 8, 745-752.
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D. Zagier. The dilogarithm function.
In: Frontiers in number theory, physics, and geometry. II, 3-65, Springer, Berlin, 2007.
Office hours
Will be announced on my homepage.
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