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Author Archives: Danny Calegari
Measure theory, topology, and the role of examples
Bill Thurston once observed that topology and measure theory are very immiscible (i.e. they don’t mix easily); this statement has always resonated with me, and I thought I would try to explain some of the (personal, psychological, and mathematical) reasons … Continue reading
Round slices of pointy objects
A regular tetrahedron (in ) can be thought of as the convex hull of four pairwise non-adjacent vertices of a regular cube. A bisecting plane parallel to a face of the cube intersects the tetrahedron in a square (one can … Continue reading
Posted in Convex geometry
Tagged Banach space, Convex geometry, Dvoretzky's theorem, L_2
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Ellipsoids and KAK
As many readers are no doubt aware, the title of this blog comes from the famous book Geometry and the Imagination by Hilbert and Cohn-Vossen (based on lectures given by Hilbert). One of the first things discussed in that book … Continue reading
Posted in Visualization
Tagged classical geometry, ellipsoids, KAK decomposition, Lie groups
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Combable functions
The purpose of this post is to discuss my recent paper with Koji Fujiwara, which will shortly appear in Ergodic Theory and Dynamical Systems, both for its own sake, and in order to motivate some open questions that I find … Continue reading
Quasimorphisms and laws
A basic reference for the background to this post is my monograph. Let be a group, and let denote the commutator subgroup. Every element of can be expressed as a product of commutators; the commutator length of an element is the … Continue reading
Groups with free subgroups (part 2)
In a previous post, I discussed some methods for showing that a given group contains a (nonabelian) free subgroup. The methods were analytic and/or dynamical, and phrased in terms of the existence (or nonexistence) of certain functions on or on … Continue reading
The (strengthened) Hanna Neumann Conjecture
A few days ago, Joel Friedman posted a paper on the arXiv purporting to give a proof of the (strengthened) Hanna Neumann conjecture, a well-known problem in geometric group theory. Simply stated, the problem is as follows. Conjecture (Hanna Neumann): … Continue reading →