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Author Archives: Danny Calegari
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
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
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 biological computation, DNA, fatgraphs, free groups, Holliday junction, scl, thermodynamics, van Kampen diagrams
4 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 Burns-Hale, distortion, Dynamics, orderable groups, quasimorphisms, Thurston stability theorem
4 Comments
The topological Cauchy-Schwarz inequality
I recently made the final edits to my paper “Positivity of the universal pairing in 3 dimensions”, written jointly with Mike Freedman and Kevin Walker, to appear in Jour. AMS. This paper is inspired by questions that arise in the … Continue reading
Posted in 3-manifolds, TQFT
Tagged 3-manifolds, Besson-Courtois-Gallot, compression body, Dijkgraaf-Witten, Hamilton, hyperbolic manifolds, JSJ decomposition, Miles Simon, minimal surface, Perelman, Ricci flow, scalar curvature, TQFT, unitary, universal pairing, volume entropy, volume inequality
3 Comments
Big mapping class groups and dynamics
Mapping class groups (also called modular groups) are of central importance in many fields of geometry. If is an oriented surface (i.e. a -manifold), the group of orientation-preserving self-homeomorphisms of is a topological group with the compact-open topology. The mapping … Continue reading
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 →