Topics in geometry 2024: negatively curved spaces
Current affairs 20.8.2024
The course begins October 28th 2024 in MaD 380.
Contents of the course
We begin with an introduction to the geometry of the hyperbolic plane, which is a prototype of a negatively curved space. We will see that, in contrast with the flat geometry of the Euclidean plane, triangles in the hyperbolic plane are thin and slim. (We will define quantitative versions of both concepts.)
We will then study geodesic Gromov-hyperbolic spaces, defined by requiring all triangles in the space to be thin or slim. Infinite simplicial trees are examples of Gromov hyperbolic spaces that are not manifolds: All edges in the following figure have length $1$ and there are exactly three edges meeting at each vertex, imagine that this construction is repeated also for the vertices at the edge of the picture to produce an infinite tree.
We will discuss compactification of Gromov hyperbolic spaces by adding a space at infinity. The space at infinity of a Gromov-hyperbolic space $X$ consists of asymptotic classes of geodesic rays (isometric embeddings of the Euclidean half-line) in $X$. This construction gives a geometric meaning for the unit disk as the boundary of the Poincaré disk model of the hyperbolic plane: The space at infinity of the hyperbolic plane is homeomorphic with the unit sphere in the Euclidean plane.
Can you see that the space at infinity of the infinite tree pictured above is a Cantor set?
Time permitting we will discuss further topics towards the end of the course.
Prerequisites
Metric spaces and topology.Lecture notes
We will follow parts of the text Geometry2024 that will be posted here.
Exercises
The exercises are included in the lecture notes. The assigned exercises for each week are listed below.
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Contact information
Jouni Parkkonen
Matematiikan ja tilastotieteen laitos
PL 35
40014 Jyväskylän yliopisto