Monthly Archives: July 2009

Brianchon-Gram-Sommerville and ideal hyperbolic Dehn invariants

A beautiful identity in Euclidean geometry is the Brianchon-Gram relation (also called the Gram-Sommerville formula, or Gram’s equation), which says the following: let be a convex polytope, and for each face of , let denote the solid angle along the … Continue reading

Posted in Euclidean Geometry | Tagged , , , , , , , , , | 4 Comments

scl, sails and surgery

I have just uploaded a paper to the arXiv, entitled “Scl, sails and surgery”. The paper discusses a connection between stable commutator length in free groups and the geometry of sails. This is an interesting example of what sometimes happens … Continue reading

Posted in Convex geometry, Groups | Tagged , , , , , , , , , | 1 Comment

van Kampen soup and thermodynamics of DNA

The development and scope of modern biology is often held out as a fantastic opportunity for mathematicians. The accumulation of vast amounts of biological data, and the development of new tools for the manipulation of biological organisms at microscopic levels … Continue reading

Posted in Biology, Dynamics, Groups | Tagged , , , , , , , | 4 Comments

Amenability of Thompson’s group F?

Geometric group theory is not a coherent and unified field of enquiry so much as a collection of overlapping methods, examples, and contexts. The most important examples of groups are those that arise in nature: free groups and fundamental groups … Continue reading

Posted in Commentary, Groups | Tagged , , , , | 12 Comments

Orderability, and groups of homeomorphisms of the disk

I have struggled for a long time (and I continue to struggle) with the following question: Question: Is the group of self-homeomorphisms of the unit disk (in the plane) that fix the boundary pointwise a left-orderable group? Recall that a … Continue reading

Posted in Dynamics, Groups | Tagged , , , , , | 4 Comments