Low Dimensional Topology

April 16, 2014

Topology blogs

Filed under: Misc. — Jesse Johnson @ 1:21 pm

Along with not writing many posts over the last year, I also haven’t been reading many math blogs. But I just stumbled across Alex Sisto’s blog, and wanted to share the link. He has a number of really nice posts related to curve complexes, mapping class groups, and even a trefoil knot complement cake. If you haven’t read it before, you should go and read it now.

By the way, if you happen to know of any other good geometry/topology blogs that aren’t in our blog roll (on the right side of the page), please feel free to include the link in a comment so I can add it.

April 4, 2014

The Thin Manifold Conference

Filed under: Misc.,Thin position — Jesse Johnson @ 11:29 am

I just wanted to point everyone’s attention to an upcoming conference The Thin Manifold, being organized by my long-time collaborators Scott Taylor and Maggy Tomova. The main theme of the conference will be thin position for knots and three-manifolds, with many of the talks focusing on the sort of hands-on, cut-and-paste geometric topology that I’ve been writing about on this blog.

There will be some travel funding available for graduate students and early career mathematicians. Before the conference, there will be graduate student workshops, led by Jessica Purcell, who has been doing a lot of very cool work on WYSIWYG geometry/topology and Alex Zupan, who has been proving a lot of nice results about thin position and bridge surfaces. The graduate student workshop is August 5-7, and the conference is August 8-10. I’m looking forward to it and hope to see you there.

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.

Conference banner
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November 19, 2013

What is the Shannon Capacity of a coloured knot?

Filed under: Knot theory,Misc. — dmoskovich @ 10:41 am

I see topological objects as natural receptacles for information. Any knot invariant is information- perhaps a knot with crossing number n is a fancy way of writing the number n, or a knot with Alexander polynomial \Delta(X) is a fancy way of carrying the information \Delta(X). A few days ago, I was reading Tom Leinster’s nice description of Shannon capacity of a graph, and I was wondering whether we could also define Shannon capacity for a knot. Avishy Carmi and I think that we can (and the knots I care about are coloured), and although the idea is rather raw I’d like to record it here, mainly for my own benefit.

For millenea, the Inca used knots in the form of quipu to communicate information. Let’s think how we might attempt to do the same. (more…)

July 8, 2013

Tangle Machines- Part 1

Filed under: Combinatorics,Misc. — dmoskovich @ 11:27 am

In today’s post, I will define tangle machines. In subsequent posts, I’ll realize them topologically and describe how we study them and more about what they mean.

To connect to what we already know, as a rough first approximation, a tangle machine is an algebraic structure obtained from taking a knot diagram coloured by a rack, then building a graph whose vertices correspond to the arcs of the diagram and whose edges correspond to crossings (the overcrossing arc is a single unit- so it “acts on” one undercrossing arc to change its colour and to convert it into another undercrossing arc). Such considerations give rise to a combinatorial diagrammatic-algebraic setup, and tangle machines are what comes from taking this setup seriously. One dream is that this setup is well-suited to modeling mutually interacting processes which satisfy a natural `conservation law’- and to move in a very applied direction of actually identifying tangle machine inside data.

To whet your appetite, below is a pretty figure illustrating a 9_{26} knot hiding inside a synthetic collection of phase transitions between anyons (an artificial and unrealistic collection; the hope is to find such things inside real-world data):

9_26 example

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Tangle Machines- Positioning claim

Filed under: Combinatorics,Knot theory,Misc.,Quantum topology — dmoskovich @ 11:09 am

Avishy Carmi and I are in the process of finalizing a preprint on what we call “tangle machines”, which are knot-like objects which store and process information. Topologically, these roughly correspond to embedded rack-coloured networks of 2-spheres connected by line segments. Tangle machines aren’t classical knots, or 2-knots, or knotted handlebodies, or virtual knots, or even w-knot. They’re a new object of study which I would like to market.

Below is my marketing strategy.

My positioning claim is:

  • Tangle machines blaze a trail to information topology.

My three supporting points are:

  1. Tangle machines pre-exist in a the sense of Plato. If you look at a knot from the perspective of information theory, you are inevitably led to their definition.
  2. Tangle machines are interesting mathematical objects with rich algebraic structure which present a plethora of new and interesting questions with information theoretic content.
  3. Tangle machines provide a language in which one might model “real-world” classical and quantum interacting processes in a new and useful way.

Next post, I’ll introduce tangle machines. Right now, I’d like to preface the discussion with a content-free pseudo-philosophical rant, which argues that different approaches to knot theory give rise to different `most natural’ objects of study.

(more…)

May 19, 2013

Flooved

Filed under: Misc. — dmoskovich @ 8:52 am

Exciting news in academic publishing!

There’s a startup company in the UK, called Flooved, who are on a mission to revolutionize scientific publishing. What sets them apart from many similar-sounding initiatives is that they seem to have a solid business model and they seem to be doing all of the right things, therefore my bet is that they are going to succeed.

What they do is to compile existing lecture notes, handouts and study-guides, and along the lines of the Open Access movement, to make them freely available online. The advantage to students is clear. The advantage to instructors is that more people read and use the material. The advantage to publishers who contribute content (are you listening, big publishing companies?) is that they get precise and useful information on how the students are using their content, and this helps them make informed decisions to put them ahead of the competition. Beyond this, the Flooved model makes education available to people worldwide, including to people who don’t have access to universities. Now, if only they could also provide assessment and accreditation…

August 27, 2012

Cornell Tribute to Bill Thurston

Filed under: Misc. — Jesse Johnson @ 1:00 pm

Cornell has created a tribute page to Bill Thurston with links to biographical information and remembrances. (Tim Riley posted this in the comments on Nathan’s post, but I wanted to make sure everyone saw it.) Thurston’s work is certainly fundamental to most of the mathematics that I think about (see, for example, this post) but just as important is what he contributed to mathematical culture and community. If you haven’t read On Proof and Progress in Mathematics, you should (and if you haven’t read it recently, you should read it again). I have always been proud to be part of a field where someone as kind, generous and selfless as Bill Thurston could become one of the leading and most prominent figures. He will be deeply missed.

 

July 15, 2012

Low dimensional topology makes good television

Filed under: Misc. — dmoskovich @ 10:28 am

Tomorrow, on Monday July 16th from 25:23 to 25:53 (i.e. July 17th from 1:23-1:53 AM), Fuji Television will screen an episode of “Takeshi Kitano presents Comaneci University Mathematics” focussing on Knot Theory! Although not exactly at a prime time slot, this is an Emmy nominated popular TV series. It will be focussed on Kouki Taniyama’s Knot Theory seminar at Waseda University, and they will have permission to upload clips to Kouki Taniyama’s homepage after the episode has been screened. This is major media exposure. One hopes that the ratings will be as high as possible, and that other TV stations in other countries will catch on to the fact that low dimensional topology makes good television.

The best exposure low dimensional topology ever got in Japan, I think, was NHK’s 2007 Special Why the 100-year-old conjecture was proven about Perelman, Geometrization, and the Poincaré Conjecture. This documentary told a compelling story to people with no mathematical background, to entertain instead of to educate. There was virtually no gossip in it (unlike media coverage of the topic in Russia, for example), and the real hero was the mathematics. At the dramatic climax, you feel like shouting out “Of course! The key missing idea is differential geometry on Alexandrov spaces!!” without necessarily knowing what any of those words mean. It’s just very good television. I couldn’t find it online (it’s in Japanese anyway, so inaccessible for most readers without a translation) because it’s copywrited material, but I did find this video which, despite being heavily edited, gives some flavour of what it was like. (more…)

April 15, 2012

Kuperberg proves that knottedness is in NP

Filed under: 3-manifolds,Combinatorics,Computation and experiment,Misc. — dmoskovich @ 8:40 am

Gil Kalai, my old Graph Theory professor at Hebrew University, and a great mathematical inspiration, who won the Rothschild Prize a few weeks ago (congratulations Gil!), wrote a very nice blog post about another massive recent result in low dimensional topology.

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