Fakultät II
Institut für Mathematik

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# Applications of Hyperbolic Geometry

## Boris Springborn

### Lectures

Mondays & Thursdays at 10:15-11:45 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 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.

### Literature

 Boris Springborn . 16.04.2021.