The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here. What are the various real life applications of topology?
user168764 asked Oct 18, 2011 at 16:36 2,977 5 5 gold badges 22 22 silver badges 25 25 bronze badges $\begingroup$ Search on "applied topology." $\endgroup$ Commented Oct 18, 2011 at 16:49 $\begingroup$ @Fredrik, never met this fact nor any of its consequences in real life. $\endgroup$ Commented Oct 18, 2011 at 17:35 $\begingroup$ For making Klein steins. $\endgroup$ Commented Oct 18, 2011 at 18:40 $\begingroup$ Computational Topology has many real life applications $\endgroup$ Commented Jan 22, 2012 at 11:40$\begingroup$ Fixed-point theorems. It has applications in diff and integro-diff and other equations. It gives the existence (or maybe the uniqueness) of the solution. $\endgroup$
Commented Aug 16, 2012 at 15:51See "Topological Insulators", an invention that takes electronics to a new phase.
54.5k 20 20 gold badges 198 198 silver badges 374 374 bronze badges answered Oct 18, 2011 at 17:14 26.2k 6 6 gold badges 70 70 silver badges 179 179 bronze badges $\begingroup$Robert Ghrist uses algebraic topology to improve sensor networks and robotics.
Twisted K-Theory is used to classify D-branes in string theory.
answered Oct 18, 2011 at 17:00 17.5k 3 3 gold badges 39 39 silver badges 87 87 bronze badges $\begingroup$ D-branes and string theory are real life now? :) $\endgroup$ Commented Oct 18, 2011 at 17:08$\begingroup$ Ghrist has also written about using "persistent homology" in image recognition software. $\endgroup$
Commented Oct 18, 2011 at 19:25 $\begingroup$
$\begingroup$ 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.) $\endgroup$
Commented Mar 22, 2015 at 14:14 $\begingroup$First, wherever you have a structure with some notion of continuity, you usually have a topology lurking in the background. You don't want to prove the same theorems over and over again in metric spaces, differential manifolds, normed vector spaces. Too many sets in every branch of mathematics 'automatically' come with a topology for topology to be ignored.
Second, continuity is a tangible notion if any mathematical notion is. What could be more important to real life than curves and other maps which are actually continuous? In most tangible situations, continuity is the first criterion that a function is reasonable. Consider configuration spaces for example: suppose you have a pendulum(1), with another pendulum(2) hooked to (1) at the end. The first pendulum sweeps out a circle, and given each point on that circle, pendulum (2) sweeps out another circle independently. The space of configurations is therefore $\mathbb \times \mathbb$, the product of two circles which forms a topological space that looks like a torus. Now, even before setting up differential equations, etc. it's immediately obvious that the path of the pendulum system should be continuous. Surprisingly, you can often prove a lot using only topology, even before you start using manifolds and other structure (like differentiability, etc).
answered Oct 18, 2011 at 17:36 Rex Butler Rex Butler 1,642 12 12 silver badges 19 19 bronze badges $\begingroup$Though really the first two apps listed below are only tangentially "topology" and more dynamics (definitely also intersect with probability, geometry, measure theory among other things) but look up the following if you wish:
(1) Collage Theorem, Fractal Compression - log onto World of Warcraft say and look around at the trees and mountains if you want to see examples of fractals.
(2) Fractal Antennas over traditional antennas.
(3) Persistant Homology (a refinement of Morse theory in topology) has proven very viable in finding patterns in large dimensional data. Recall if you have just two variables and measurements with errors, it is quite often useful to find the best 1-dimensional manifold (hehe curve) of given type that fits the data. Well if you have 10000 variables, and a bunch of data, picture understanding the basic "shape" of a best fit manifold ("nice shape") to the data and this is very very roughly what this is about. As you can guess the solution might involve geometry and topology.. :)
(4) As already mentioned nicely above, movement planning problems - given a robot how to move it from point A to point B efficiently without knocking everything over and/or falling on its face - think about how much coordination your arms and legs have to have to dance a ballet dance for example and think of how you would program a robot to do it - you'll quickly come to the conclusion that geometry and topology have something to do with it!
(5) The geometric understanding of spacetime afforded by Einstein's general relativity is even applied as before the clocks in GPS sattelites were corrected for said effects, the accuracy of GPS was less than optimal.
(6) Usually the question of "real world applications" just means that the discussed bit of math doesnt belong to the batch of math that the questioner uses everyday. Over time, I don't think I have seen anything stay "useless" long - number theory stayed "useless" for 2000 years till computers and security and the demand for quick algorithm speeds suddenly made it very useful! Like someone once said when discussing the math behind special effects and the gaming/movie industry - around 10 years ago, math dispensed with the "real world". I am not really sure what "real world application" really means anymore.
(7) I am not sure this is real world, but political scientists and economists often use Morse theory to discuss the stability of game theory and market equilibria. They have used fixed point theory from topology such as Brouwer's fixed point theorem and Kakutani's fixed point theorem for a long time now and in the last 10 years Morse theory and more refined parts of topology have come into play in these areas.
(8) Threshold complexes from topology were used to disprove a computer science conjecture about the complexity of certain algorithms. Complexity of algorithms has been studied using topological techniques including the Borsuk-Ulam theorem.
281k 27 27 gold badges 310 310 silver badges 591 591 bronze badges answered Jan 22, 2012 at 0:00 Jonathan Pakianathan Jonathan Pakianathan 251 3 3 silver badges 2 2 bronze badges $\begingroup$Topology helps understanding the molecular structures. See this book, When Topology Meets Chemistry: A Topological Look at Molecular Chirality written by Erica Flapan. I skimmed a few chapters of the book and it was very interesting.
answered Jan 22, 2012 at 1:43 Karatuğ Ozan Bircan Karatuğ Ozan Bircan 4,339 1 1 gold badge 26 26 silver badges 53 53 bronze badges $\begingroup$Do a web search on "Knots and DNA". In particular, look here.
In the 1970s I gave up teaching undergraduates about homology in favour of knot theory, as: it was more fun; it was related to some nice group theory; the students could see immediate problems, such as how do really know the trefoil knot is knotted; many nice computations to do; there is wonderful history in knots, as possibly the oldest form of applied geometry/topology; and it led me into giving popular lectures on the subject, which led to all sorts of things.
Another area which is not taught so much is the Hausdorff metric. I have given this in a second year analysis course and it is more fun than uniform convergence. I quoted a completeness theorems; there are lots of nice applications to fractals, which links to an area of public knowledge. I even asked for a brief essay as a test on "The importance of fractals" all to be derived from books or the web. Some students used fractal programs in their answer.
answered Jun 25, 2013 at 15:02 Ronnie Brown Ronnie Brown 15.5k 2 2 gold badges 43 43 silver badges 53 53 bronze badges$\begingroup$ Sorry I'm a bit late for the party, but I found your answer to be quite interesting. I'm a graduate student yet I must admit that I know very little about the Hausdorff metric apart from a couple of times I saw it used it in measure-related theorems. Do you have some suggestions on where I can start reading more about it? It'd be nice to see some connections to the topics you mentioned as well. $\endgroup$
Commented Feb 13, 2021 at 8:01$\begingroup$ There asre lots of books at various levels on fractals and chaos, and popular computer programmes. I found it fun to teach, abd asked for a test stduents to write a short essway on the importance of fractals. One can assume competeness, where necessary, without proof, byt wwith understanding its meanong. $\endgroup$
Commented Feb 20, 2021 at 9:44$\begingroup$ I've just come across the paper "Geometry and the imaginagination" by John Conway, Bill Thurston, et al/ Find it on a search google. Or try Lecture notes published on …, 1991 - pdf-drive.com $\endgroup$
Commented Feb 20, 2021 at 9:51$\begingroup$ Thank you for mentioning the paper, I'll surely have a look! If you don't mind, may I ask if you have any specific book recommendation for a person of my background? I'm comfortable with graduate-level analysis in general and know some basic geometric measure theory. $\endgroup$
Commented Feb 20, 2021 at 16:30$\begingroup$ Try my book Topology and Groupoids. Downloafabe from my web site: groupoids.org.uk . I do not know if it meets your back ground abd interests. . $\endgroup$
Commented Feb 27, 2021 at 21:38 $\begingroup$Search in the journal of Topology and its Applications gives a paper on titled Inverse limit spaces arising from problems in economics
answered Jan 22, 2012 at 2:10 Sniper Clown Sniper Clown 1,208 2 2 gold badges 19 19 silver badges 36 36 bronze badges $\begingroup$Abstract
In this paper we use tools from topology and dynamical systems to analyze the structure of solutions to implicitly defined equations that arise in economic theory, specifically in the study of so-called “backward dynamics”. For this purpose we use inverse limit spaces and shift homeomorphisms to describe solutions which are typical in that they are likely to be observed in future time. These predicted solutions corresponds to attractors in an inverse limit space under the shift homeomorphism(s).
This is something I can think off my head:
The fixed point theorems in topology are very useful. Here's one account of how the problem was formulated:
A physicist wanted to consider a flat plate on which one part of water and another part of oil are mixed together. He asked whether there is any point that doesn't move when mixing!
The answer is YES. It boils down to asking, does the space $\$ have the fixed point property or not. And, indeed it has!
In fact, there is also a point that doesn't move when you stir a glass of milk. Interesting; Strange but true!!
As pointed out in the comments, this point need not be invariant at all times. All that is asserted is, there is always one point that does not move at every given time $'t'$,
And, I'd suggest you read the book, "A First Course in Topology", by MCleary.
Hope this helps!
answered Jan 22, 2012 at 0:18 user21436 user21436$\begingroup$ Nice, but there is no search result about "A First Course in Topology" by MClearsky. $\endgroup$
Commented Jan 22, 2012 at 1:08 $\begingroup$ Probably McCleary? $\endgroup$ Commented Jan 22, 2012 at 1:15 $\begingroup$ Oh, fixed @Tim, Thanks $\endgroup$ Commented Jan 22, 2012 at 1:21$\begingroup$ "there is also a point that doesn't move when you stir a glass of milk." My understanding was that at any given time t, there is always at least one point that is in its original position, but that there does not necessarily have to be a single point invariant for all times t. $\endgroup$
Commented Jan 22, 2012 at 2:25$\begingroup$ A physicist wanted to consider a flat plate on which. There seems to be no lower bound to the degree of plausibility of the fairy tales proposed as concrete justifications for the activity called mathematics. $\endgroup$
Commented Jan 22, 2012 at 11:31 $\begingroup$There are applications of Topology in Biology: Topology in Molecular Biology.
answered Aug 16, 2012 at 15:28 2,915 3 3 gold badges 28 28 silver badges 34 34 bronze badges $\begingroup$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.
