Low Dimensional Topology

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…)

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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

(more…)

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.

(more…)

March 5, 2012

Rethinking the basics

Filed under: Misc.,Quantum topology,Triangulations — dmoskovich @ 3:36 am

Some nights, one gazes up at the stars, and thinks about philosophy. Who are we? What is the meaning of life? What is reality? What are manifolds really?

This morning, I looked at Poincaré’s original definition in Papers on Topology: Analysis Situs and Its Five Supplements, translated by John Stillwell. His original definition was pretty-much that a manifold is a quotient of \mathbb{R}^n by a properly discontinuous group action, that group being his original fundamental group. Implicitly, his smooth, PL, and topological categories were all the same thing (indeed true for 1-manifolds, and for dimensions 2 and 3 PL and smooth categories still “coincide” in a sense that can be made fully precise); nowadays we understand that the situation is more subtle. But I’m still not sure that I understand what a manifold is- what it really is.

In some non-mathematical, philosophical (theological?) sense, I believe that both smooth and PL manifolds actually exist, in the sense that natural numbers exist, and tangles exist. Our clumsy formal definitions are attempts at describing something that is actually out there, as the Peano axioms describe the natural numbers. I also believe that Physics is a guide to Mathematics, because things that really exist might also be observed… so ideas from Physics (topological invariants defined by means of path integrals) ought to be taken very seriously, and it is my irrational belief that these will eventually turn out to be the most fundamental invariants in some precise mathematical sense.

It is fascinating to me, then, that input from physics seems to be leading towards a fundamental rethink of the basic definitions of smooth and PL manifolds. I feel like we had some sub-optimal definitions, which we worked with for sociological reasons (definitions are made by people, and people are not perfect), and maybe in the not too distant future there will be a chance to put more convenient definitions in place. Maybe the real world (physics) will force it on us. Let me tell you, then, about some of the papers I’ve been (casually) flicking through recently (the one I’m most excited about is Kirillov’s On piecewise linear cell decompositions). (more…)

February 12, 2012

Math 2.0: Discuss

Filed under: Misc. — Jesse Johnson @ 9:24 pm

Scott Morrison (of Secret Blogging Seminar) and Andrew Stacey (of the nLab) have created a discussion forum called Math 2.0 to discuss the future of mathematics publishing: Should journals continue to exist? If so what should they be like? What roles will other types of publishing (arXiv, blogs, wikis, MathOverflow, etc.) play in the future landscape? Clearly, this is an important conversation to have – The major role that journals still play (as I see it) is to provide non-experts with a means of assessing and comparing mathematicians, areas within mathematics and even mathematics relative to other fields. Hiring and tenure decisions and agency funding decisions, to name just two, require a system of vetting (to borrow a term from politics) that administrators can understand and believe they can trust. The one strength of the current system is that it has a sense of legitimacy (at least to outsiders.) It seems that we, as a community, now have an opportunity to change the system, or perhaps even create a new one.  I hope that the discussions on Math 2.0 will help ensure that the new system is fair and effective, but also a system that will be deemed legitimate by non-mathematicians.

January 30, 2012

Topology on google+

Filed under: Misc. — Jesse Johnson @ 2:53 pm

I recently discovered that Siddhartha Gadgil is using google+ as a sort of topology (mini?)blog. His posts are all public, so you don’t need a google account to view them. He currently has a number of videos describing Stalling’s topological proof of Grushko’s Theorem: Part 1, Part 2, Part 3.    (I don’t know how to link to a specific post, rather than to the whole stream.) Are you using google+ as a blogging platform? Do you know of any other topologists who are?  If so, leave a comment. In fact, if you’ve discovered any new topology blogs that aren’t on our (outdated) blogroll, feel free to post a comment about that as well.

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