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 Bing’s wild involution
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 Schläfli – for lush, voluminous polyhedra
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 Taut foliations and positive forms
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 Mapping class groups: the next generation
 Groups quasiisometric to planes
 Div, grad, curl and all this
 A tale of two arithmetic lattices
 3manifolds everywhere
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 Kähler manifolds and groups, part 2
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 Scharlemann on Schoenflies
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Category Archives: 3manifolds
Bing’s wild involution
An embedded circle separates the sphere into two connected components; this is the Jordan curve theorem. A strengthening of this fact, called the JordanSchoenflies theorem, says that the two components are disks; i.e. every embedded circle in the sphere bounds a … Continue reading
Posted in 3manifolds, Uncategorized
Tagged Alexander horned sphere, Bing, wild involution
4 Comments
Schläfli – for lush, voluminous polyhedra
Because of some turbulence in real life (™) over the last few months, it’s been somewhat hard to concentrate on research. Fortunately the obligation of teaching sometimes exerts the right sort of psychological pressure to keep my mind on mathematics in short bursts. Continue reading
Mr Spock complexes (after Aitchison)
The recent passing of Leonard Nimoy prompts me to recall a lesserknown connection between the great man and the theory of (cusped) hyperbolic 3manifolds, observed by my friend and former mentor Iain Aitchison. In particular, I am moved to give a brief presentation … Continue reading
Taut foliations and positive forms
This week I visited Washington University in St. Louis to give a colloquium, and caught up with a couple of my old foliations friends, namely Rachel Roberts and Larry Conlon. Actually, I had caught up with Rachel (to some extent) … Continue reading
Groups quasiisometric to planes
I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the … Continue reading
Div, grad, curl and all this
The title of this post is a nod to the excellent and wellknown Div, grad, curl and all that by Harry Schey (and perhaps also to the lesserknown sequel to one of the more consoling histories of Great Britain), and the purpose … Continue reading
Posted in 3manifolds, Riemannian geometry
Tagged curl, div, exposition, grad, Riemannian geometry, vector field
12 Comments
3manifolds everywhere
When I started in graduate school, I was very interested in 3manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics … Continue reading
Posted in 3manifolds, Groups, Hyperbolic geometry
Tagged 3manifolds, acylindrical, quasiconvex group, Random groups, Sierpinski carpet
9 Comments
Scharlemann on Schoenflies
Yesterday and today Marty Scharlemann gave two talks on the Schoenflies Conjecture, one of the great open problems in low dimensional topology. These talks were very clear and inspiring, and I thought it would be useful to summarize what Marty … Continue reading