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Fortgeschrittene Themen der Differentialgeometrie
Sommer 2021

Applications of Hyperbolic Geometry

Boris Springborn

Lectures

Mondays & Thursdays at 10:15-11:45
  • Zoom Link for first lecture: tba
  • Meeting ID tba
  • Passcode tba
First lecture on 12 April

Formalities

This module, Fortgeschrittene Themen der Differentialgeometrie, can also be counted as Geometry III or Differential Geometry III.

The lectures will be in English or German, depending on the audience.

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 . 10.04.2021.