Category Archives: Surfaces

Agol’s Virtual Haken Theorem (part 1)

I am in Paris attending a workshop at the IHP where Ian Agol has just given the first of three talks outlining his proof of the Virtual Haken Conjecture and Virtual Fibration Conjecture in 3-manifold topology (hat tip to Henry … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , | 4 Comments

The Hall-Witt identity

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which … Continue reading

Posted in Groups, Lie groups, Surfaces, Visualization | Tagged , , , | 1 Comment

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem. 1. Triangles in Hyperbolic Space Suppose we are given … Continue reading

Posted in Euclidean Geometry, Geometric structures, Hyperbolic geometry, Surfaces, Uncategorized | 1 Comment

Polygonal words

Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov: Question(Gromov): Let be a one-ended word-hyperbolic group. … Continue reading

Posted in Groups, Surfaces | Tagged , , , , , , , , | 6 Comments

Minimal laminations with leaves of different conformal types

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it … Continue reading

Posted in Complex analysis, Surfaces | Tagged , , , , , , | 8 Comments

Bridgeman’s orthospectrum identity

Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The -dimensional case is (in my opinion) the most beautiful, … Continue reading

Posted in Hyperbolic geometry, Special functions, Surfaces | Tagged , , , , | 6 Comments

Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface

Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the … Continue reading

Posted in Complex analysis, Euclidean Geometry, Surfaces | Tagged , , , , , , , | 4 Comments