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Author Archives: Danny Calegari
Characteristic classes of foliations
I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes … Continue reading
Filling geodesics and hyperbolic complements
Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact … Continue reading
Quasigeodesic flows on hyperbolic 3-manifolds
My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress … Continue reading
Laying train tracks
This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether … Continue reading
Posted in Ergodic Theory, Euclidean Geometry
Tagged central limit theorem, local limit theorem, Markov chain, tiling, train tracks
19 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 commutators, gropes, Hall-Witt identity, visualization
1 Comment
Ziggurats and the Slippery Conjecture
A couple of months ago I discussed a method to reduce a dynamical problem (computing the maximal rotation number of a prescribed element in a free group given the rotation numbers of the generators) to a purely combinatorial one. Now Alden Walker … Continue reading
Posted in Dynamics
Tagged Arnol'd tongues, combinatorics, Rigidity, rotation number, ziggurats
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Zonohedra and the Sylvester-Gallai theorem
When I was in Melbourne recently, I spent some time browsing through a copy of “Twelve Geometric Essays” by Harold Coxeter in the (small) library at AMSI. One of these essays was entitled “The classification of zonohedra by means of … Continue reading
Posted in Polyhedra, Projective geometry
Tagged Coxeter, projective plane, Sylvester-Gallai theorem, zonohedra
4 Comments
