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Monthly Archives: June 2009
The topological CauchySchwarz 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 3manifolds, TQFT
Tagged 3manifolds, BessonCourtoisGallot, compression body, DijkgraafWitten, Hamilton, hyperbolic manifolds, JSJ decomposition, Miles Simon, minimal surface, Perelman, Ricci flow, scalar curvature, TQFT, unitary, universal pairing, volume entropy, volume inequality
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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 orientationpreserving selfhomeomorphisms of is a topological group with the compactopen topology. The mapping … Continue reading
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 nonadjacent 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 CohnVossen (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