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Fortgeschrittene Themen der Differentialgeometrie Sommer 2021
Applications of Hyperbolic Geometry
Lectures
Mondays & Thursdays at 10:1511:45 via Zoom (see blog)
Since it turned out that everybody speaks German, the lectures are in German.
Blog
After the lectures, I will post the blackboards in the course blog, which is also a good place for comments, questions and discussion. Please write me an email with your personal data (name, Studiengang, or whatever is appropriate) to register.
Formalities
This module,
Fortgeschrittene Themen der Differentialgeometrie,
can also be counted as
Geometry III
or
Differential Geometry III.
Contents
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
of Andrey
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
the Farey
triangulation, and how well a number can be approximated depends
on how far these geodesics stay away from
the Ford
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 threedimensional 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 3space.
Literature
 Hyperbolic geometry

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 1138.

Cannon, Floyd, Kenyon & Parry,
Hyperbolic Geometry. In S. Levi (ed.), Flavors
of Geometry, MSRI Publications 31, Cambridge University
Press, 1997.

J. Milnor, Hyperbolic Geometry: The first 150
years. Bull. Amer. Math. Soc. (N.S.) 6,
(1982), no. 1, 924.

F. Bonahon, LowDimensional 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.

J. Conway,
The
Sensual Quadratic Form, MAA, 1997.

L. R. Ford, Fractions, Amer. Math. Monthly 45 (1938), 586601.

A. Hatcher, Topology
of numbers.

B. Springborn, The
hyperbolic geometry of Markov's theorem on Diophantine
approximation and quadratic forms, Enseign. Math.
63:34 (2017), 333373.
 Discrete conformal maps

A. I. Bobenko, U. Pinkall,
B. Springborn, Discrete
conformal maps and ideal hyperbolic
polyhedra, Geom. Topol. 19:4 (2015), 21552215.

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), 63108.
