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What are the various real life applications of topology?

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17 Answers 17

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Robert Ghrist uses algebraic topology to improve sensor networks and robotics.


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From wikipatents:

Woven endless belt of a spliceless and Mobius strip construction

http://www.wikipatents.com/US-Patent-3991631/woven-endless-belt-of-a-spliceless-and-mobius-strip-construction LINK DEAD

patents.com link

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Link is dead. :( Long live the link! –  SpamIAm Jan 27 at 20:10
It's better practice to post the info rather than just a link, then you can never have a dead link and people can always search for the latest home of that info. In this case, US Patent #3991631: "Woven endless belt of a spliceless and Mobius strip construction." (At least in this case the link URL contains this info.) –  Nate C-K Mar 22 at 14:14


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(2) Fractal Antennas over traditional antennas.







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This is something I can think off my head:







Hope this helps!

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Topology also has applications within computer science.

  • Directed algebraic topology is a branch of algebraic topology that has applications in concurrency theory when trying to avoid and resolve deadlocks and starvation. See for example here.
  • Topological data analysis is an alternative to standard data mining, which allows one to infer global and structural properties about data. Wikipedia has more.
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Check out the portal Applied Topology, which arose from Gunnar Carlsson's research group in Stanford. There are many applications areas mentioned with relations to statistics, data-mining, biology etc.

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I'm not sure if this is a real-world application, but topological methods can be used for a formal proof that a system of real (possibly nonlinear) equations has solution.

Assume that $f: K\to\mathbb{R}^n$ is continuous and you want to prove that $f^{-1}(0)$ is nonempty. If, for example, $K$ is an $n$-manifold, then a nonzero degree of $f/|f|: \partial K\to S^{n-1}$ yields a zero of $f$ in $K$. For higher-dimensional domains, a generalization is described in this paper.

Similarly, the much more complicated question of existence of a global solution of an ODE can be often resolved by topological methods. An ODE solution is usualy the fixed point of a suitable operator (solution of an inital-value problem is a fixed point of the Picard-Lindolf operator, for example) and the existence of a fixed point of $T: X\to X$ acting on a Banach space can be proved by showing that the Leray-Schauder degree associated to $T$ and some set $U\subseteq X$ is nonzero. This is a very nice and readable introductory book on this topic.

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Kuratowski's Theorem characterizing planar graphs (http://en.wikipedia.org/wiki/Kuratowski%27s_theorem) has application to circuit boards where certain nodes must be connected but without any edge crossings.

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Topology is at least partially built into human intuition because it talks about invariants - general properties and classification independent of fine details - exactly what humans are best at!

There are many real-life examples, for instance, the hairy ball theorem is encountered every time you wonder how come we can't have a map of the earth without having two poles where latitude is undefined. Or when trying to comb your hair. People who make clothes know very well that the more holes you need (sleeves,...), the more additional seams you need after you cut the main piece of cloth. The minimal number of seams is basically the first betti number. Making pants is fundamentally different from making a sweater, because the number of holes is different. Origami (or wrapping presents) falls into the same category.

Knots are very important in everyday life, even if you are not a mountain climber. You subconsciously realize it is important if something is a knot or not. If you tie your shoelaces correctly, you didn't make a true knot, so you can pull a strand and it unties. However, if you mess it up, you made a knot (and lost a few minutes to undo it). The same goes for earphones. While it may not be critically important to know how to classify knots and how to define all the complicated invariants seen in knot theory, on some basic level, you do grasp the concept of something being equivalent and just arranged differently, compared to something being fundamentally different. In one case, there is only a continuous rearrangement of the rope that separates you from falling to your death, in the other case, cutting is the only way.

Another example is map coloring: even if you don't make maps, as a child you probably tried coloring something so that adjacent fields don't share the same color. You probably even noticed that if you made a closed squiggle with a single stroke, an alternating pattern of two colors is enough. But nevertheless, it took mathematicians quite a long time (and a computer) to finally prove that you need at most 4 colours. That's nothing else but topology.

Of course, there is a thin line between topology (at least "practical" topology in 2D and 3D) and geometry, so there is an entire world of everyday problems where you have at least a hint of both.

In material science, the examples are: magnetic skyrmions and related solitons, which will help fit more data on a hard drive; liquid crystals have defects that are governed by quite complex topological rules - whether you want to avoid defects or control them, you need to know the rules; DNA knotting and topological insulators were already mentioned by other answers; topology of neural networks is a way of making sense of the mess of data you acquire in brain research; other topology-related physics questions are less "applicative" and could be regarded as purely academic: topology of curved space-time, study of knotted vortices in liquids, helicity of magnetic fields...

It depends on what you mean by "using" topology. In everyday life, you are dealing with some rudimentary topological notions subconsciusly, without actually performing any real math. In physical sciences, topology is currently sort of a hype: it has seen a high increase in research and publication volume - it will definitely yield new useful stuff, but there is also a lot of papers that are just there because it's interesting to look at something from topological perspective.

Both use cases actually use a very small subset of what mathematicians call topology. So for them, topology encountered in physics on the academic level isn't much better than counting poles on a globe.

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This sounds to me a bit exaggerated, like a grant proposal :-) Sure all these things you talk about have been here long before the idea of topology and don't need it. (Moreover, we can have a "map" of the earth with only one point missing) –  Peter Franek Jan 28 at 12:50
Well, it's a discussion thread :) Practically speaking, there's a question what actually constitutes "using topology" (and, usage by common people or by scientists?). Of course, you can have a single double pole. Or even 4 half-poles... But you can't have no poles. –  orion Jan 28 at 12:55
Ok, I'm just not completely sure about those half-poles.. –  Peter Franek Jan 28 at 13:11
Line fields (vectors with no arrows) allow half defects. Representative image and paper: rspa.royalsocietypublishing.org/content/royprsa/early/2012/02/… dx.doi.org/10.1007/s00396-010-2367-7 –  orion Jan 28 at 13:30

protected by Asaf Karagila Mar 19 '14 at 19:52


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