Low Dimensional Topology

November 26, 2013

What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”.  It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

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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:

June 9, 2008

Folding proteins for fun and profit

Filed under: Metric geometry,Misc. — Jesse Johnson @ 10:32 am

It appears that some crafty biologists have figured out a way to trick unsuspecting internet users into helping them find minimal energy embeddings of complex proteins in R^3 (i.e. protein folding). The game fold it allows players to manipulate proteins and scores them based on how efficient an embedding they can find. The best solutions are then recorded by the main site. (There’s also a nice tutorial for players who don’t know any biology.)

Recall that a protein is a chain of amino acids linked by single-bonded carbon atoms that allow the joints to rotate. Different angles will determine different distances between atoms in the protein, so different embeddings have different amounts of potential energy. In nature, the protein will twist along the joints to take on an embedding that minimizes the energy. Scientists have figured out how to read off the sequence of amino acids in a protein, but figuring out the lowest energy embedding is not so easy, since there are infinitely many possible configurations.

There have been computer simulations of protein folding for a number of years now, but this only solves part of the problem – all the computer simulations may just just be finding local minima and missing the actual solutions. Fold it takes advantage of some of the things the human mind still does better than a computer. (I think this is called crowd sourcing.) I wonder if there are any math problems that could benefit from a similar campaign. One could probably get ropelength estimates this way, but they would still just be estimates rather than a proof.

December 5, 2007

Conformal dimensions and Cantor sets

Filed under: Metric geometry — Jesse Johnson @ 7:08 pm

I don’t know if metric geometry qualifies as low dimensional topology, but today’s subject happens to be low dimensional (usually somewhere between one and two) and in some sense has more to do with topology than geometry. I don’t normally look at the posts over in the .MG section of the arXiv, but I went seeking it out after hearing about it from the author (a grad. student here at Yale). Since I’m new to metric geometry, there’s a reasonable chance some of the details of my description will be off, but here goes:

Given a metric space, one can define its conformal dimension by considering all metric spaces that are quasi-conformal to the original space and taking the infimum of their Hausdorff dimensions. This is useful, for example, when looking at limit sets of discrete groups of hyperbolic isometries, since the metric on such a set is only defined up to conformal maps of the boundary sphere. As one might guess, the conformal dimension is rather hard to calculate. It also seems to have more to do with the topology of the set than its geometry.

John Mackay’s recent preprint [1] demonstrates that the conformal dimension will be strictly greater than one whenever two (essentially topological) conditions are satisfied: First, the space must be N-doubling, for some N, which means that every metric ball in the set is covered by N metric balls of half the original radius. Second, it must be annulus linearly connected, which is a metric analogue to being locally connected with no local cut points. (A cut point is a point such that in a small neighborhood, removing the point makes the neighborhood disconnected.) For the case when the set is the limit set of discrete group of hyperbolic isometries, the local cut point appears to correspond to the group virtually splitting over an elementary group. As a corollary of the main theorem, the author proves that if such a group does not virtually split over an elementary group then its limit set has conformal dimension strictly greater than one.

The proof begins with a result of P. Tukia that allows one to straighten an arc in a doubling metric space. The “straightened” arc has the property that the diameter of every small sub arc is not too much larger than the distance between its endpoints. Because the set has no local cut points, one can then find a second arc that follows parallel to the first one, then straighten it using the theorem. Then one can find two more parallel arcs, then four, etc. These sets limit to a set that looks like a Cantor set cross an interval.  Such a set is known to have conformal dimension bounded away from one and is embedded in the original set, so the conformal dimension of the original set is also bounded below.

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