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Fortgeschrittene Themen der Differentialgeometrie
Applications of Hyperbolic Geometry
Mondays & Thursdays at 10:15-11:45
First lecture on 12 April
Zoom Link for first lecture: tba
Meeting ID tba
Fortgeschrittene Themen der Differentialgeometrie,
can also be counted as
Differential Geometry III.
The lectures will be in English or German, depending on the audience.
The course is mainly about two topics where hyperbolic geometry is useful for something else.
The hyperbolic geometry of numbers
Some irrational numbers are better approximated by rational numbers
than others. The very worst approximable irrational number turns out
to be the golden ratio (1+√5)/2, followed by √2, and
(9+√221)/10 on third place. More precisely, each of these
numbers represents a class of equally badly approximable
irrationals. The very worst classes form an infinite sequence, the
study of which goes back to work
Markov on quadratic forms. Classically, the principal tool in this
area of Diophantine approximation are continued fractions. We will
treat this subject using hyperbolic geometry. From this point of view,
continued fractions describe the paths of geodesics in
triangulation, and how well a number can be approximated depends
on how far these geodesics stay away from
circles, which are really horocycles.
Discrete conformal maps
The Uniformization Theorem of complex analysis says that every compact
Riemann surface is conformally equivalent to quotient of the sphere,
the Euclidean plane, or the hyperbolic plane by the action of a
discrete group of isometries. A discrete version of this theorem deals
with polyhedral surfaces instead of Riemann surfaces. This theory is
best understood using three-dimensional hyperbolic geometry. The
discrete uniformization problem for polyhedral surfaces turns out to
be equivalent to the following problem: to realize a given complete
hyperbolic metric on a punctured surface as a convex polyhedral
surface with ideal vertices in hyperbolic 3-space.
- Hyperbolic geometry
D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov. Geometry
of spaces of constant curvature. In: E. B. Vinberg
II. Encyclopedia of Mathematical Sciences 29. Springer,
Berlin, 1993. Pages 1-138.
Cannon, Floyd, Kenyon & Parry,
Hyperbolic Geometry. In S. Levi (ed.), Flavors
of Geometry, MSRI Publications 31, Cambridge University
J. Milnor, Hyperbolic Geometry: The first 150
years. Bull. Amer. Math. Soc. (N.S.) 6,
(1982), no. 1, 9-24.
F. Bonahon, Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots, AMS, 2009
- Quadratic forms, Diophantine approximation and the hyperbolic geometry of numbers
M. Aigner, Markov's theorem and 100 years of the uniqueness conjecture: a mathematical journey from irrational numbers to perfect matchings, Springer, 2013.
Sensual Quadratic Form, MAA, 1997.
L. R. Ford, Fractions, Amer. Math. Monthly 45 (1938), 586-601.
A. Hatcher, Topology
B. Springborn, The
hyperbolic geometry of Markov's theorem on Diophantine
approximation and quadratic forms, Enseign. Math.
63:3-4 (2017), 333-373.
- Discrete conformal maps
A. I. Bobenko, U. Pinkall,
B. Springborn, Discrete
conformal maps and ideal hyperbolic
polyhedra, Geom. Topol. 19:4 (2015), 2155--2215.
K. Crane, Conformal geometry of simplicial surfaces, in: Proceedings of Symposia in
Applied Mathematics, AMS, 2020.
B. Springborn, P. Schröder, U. Pinkall, Conformal equivalence of
triangle meshes, ACM Trans. Graph., 27:3, 2008.
B. Springborn, Ideal hyperbolic polyhedra and discrete uniformization, Discrete Comput. Geom. 64:1 (2020), 63--108.