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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?

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@Mariano: But that's not what the theorem says. You are probably not (and might never be) at one of those two antipodal points. –  Robert Israel Oct 18 '11 at 18:25
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Topology is useful to mathematics as a whole. Mathematics is useful to humanity. By transitivity, Topology is useful to humanity. –  Austin Mohr Oct 18 '11 at 18:41
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"Just because the holy grail doesn't keep coffee hot while you drink from it doesn't mean it isn't important" ars.userfriendly.org/cartoons/?id=20070105 –  user5137 Oct 18 '11 at 18:54
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@AustinMohr: "is useful to" is unlikely to be a transitive relation. –  Grumpy Parsnip Oct 18 '11 at 19:27
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Knowledge of topology is often crucial for passing your PhD qualifying exams in math. You do want your PhD, don't you? –  Apprentice Queue Jan 22 '12 at 2:20

12 Answers 12

up vote 21 down vote accepted

See "Topological Insulators", an invention that takes electronics to a new phase.

http://www.physics.upenn.edu/~kane/pubs/p69.pdf

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

Twisted K-Theory is used to classify D-branes in string theory.

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D-branes and string theory are real life now? :) –  Mariano Suárez-Alvarez Oct 18 '11 at 17:08
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Ghrist has also written about using "persistent homology" in image recognition software. –  Grumpy Parsnip Oct 18 '11 at 19:25

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 $\mathbb{T} \times \mathbb{T}$, 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).

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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.

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Search in the journal of Topology and its Applications gives a paper on titled Inverse limit spaces arising from problems in economics

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).

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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.

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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 $\{(x,y)|x^2+y^2 \leq r\}$ 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!

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Nice, but there is no search result about "A First Course in Topology" by MClearsky. –  Tim Jan 22 '12 at 1:08
    
Probably McCleary? –  Tim Jan 22 '12 at 1:15
    
Oh, fixed @Tim, Thanks –  user21436 Jan 22 '12 at 1:21
    
"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. –  Rachel Jan 22 '12 at 2:25
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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. –  Did Jan 22 '12 at 11:31

There are applications of Topology in Biology: Topology in Molecular Biology.

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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.

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This semester I have a professor teaching a class called Topology with Applications about this. I am taking this class simply because, like you, I was wondering how useful formal knowledge about the differences (or lack thereof) between donuts and coffee mugs is in contexts outside the classroom.

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Rob Ghrist's Page: Has lots of great preprints on sensor networks, geometric and topological robotics, applied computational homology, and dynamical systems. Especially, see his preprints on expository papers. Much of the material in Topology and its Applications on robotics is based on Rob Ghrist's work.

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