Low Dimensional Topology

August 22, 2012

Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

October 29, 2009

Which knotted objects are worthy of study?

Filed under: Knot theory,Misc.,Pedagogy — dmoskovich @ 2:45 pm

Some people seem to rejoice in knotiness. To non-topologists, it’s not clear why anyone would care about even a plain old knot in S^3 (or a long knot), but to us it’s the most natural thing in the world. To them it would seem to specific, too specialized, not really interesting; but we know that they are wrong, right?
But what then, about links? High dimensional knots? Tangles? Braids? High dimensional links? Homotopy links? I’m sure we were a bit skeptical about the usefulness of these when we first saw them, but now we can just about accept them.
What about the next step? Knots and links in arbitrary manifolds? Singular knots? And then what about virtual knots? free knots? coloured knots? knotted trivalent graphs? What about these new objects of study like knotted handlebodies? Turaev’s topology of words (knotted words)?
How does one decide such a topic is interesting… why and when is extending a result about links in the 3-sphere to higher dimensional stuff or stuff in strange manifolds interesting? How does one become interested in it?
I know that I’m still a bit skeptical about virtual knots, for instance. But I’ve come to accept knotted trivalent graphs as natural… for rather strange reasons. How about all of you?

October 4, 2009

Communicating Topology in the 21st Century

Filed under: Misc.,Pedagogy — Nathan Dunfield @ 1:45 pm

Because of the importance of pictures in low-dimensional topology, communicating electronically with with collaborators, students, etc., has some special challenges. (Not that other mathematicians have it easy — I’d hate to have send lots and lots of equations via email.)

Here’s some useful tools/ideas for dealing with this, some of which I use myself, and others which I’ve only heard about.

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September 18, 2009

An open source mathematics book

Filed under: Pedagogy — Nathan Dunfield @ 2:13 am

Number theorist Emmanuel Kowalski has an interesting post about a truly open source math book on algebraic stacks. Not only can you download the entire 1302(?!) page book as a PDF file, you can get the complete LaTeX source files, and the whole thing is kept under version control so people can submit changes, etc.

I’ve been thinking that a similar approach would be good for textbooks. When I teach a course, I’m often frustrated by being unable to find a text that has everything I need. Or I do find such a text, but it’s poorly written in places, or aimed too high or low for my particular students. Or maybe there are theorems in the text that I’d like to assign as homework instead of lecturing on them. In such situations, it would be great if there was a whole collection of open-source textbooks that I could cut and paste from, massage the result a bit, and end up with something closer to the “perfect” text for a course.

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September 4, 2009

Reader survey: graduate courses in topology and geometry

Filed under: Pedagogy — Nathan Dunfield @ 4:37 pm

In my department, we’re considering whether we have too many basic graduate courses, by which I mean courses with (mostly) fixed syllabi aimed at first and second year graduate students, as opposed to advanced topics courses which never cover the same thing twice. In geometry/topology, we have not less than 11 such one-semester courses, of which two arguably belong more to algebra:

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September 3, 2009

Topology fun and games

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Pedagogy — Nathan Dunfield @ 10:21 am

Jeff Weeks, the author of SnapPea, has written some new educational games this year that are highly worth trying out:

  • A new version of his classic Torus Games for iPhone and iTouch.
  • A new set of Hyperbolic Games, including a maze, pool (very challenging), and soduku. (For OS X and Windows)

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