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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?

- Applications of the Fibonacci sequence
- Factorial of a matrix: what could be the use of it?
- (Possible) application of Sarason interpolation theorem
- Can you provide me historical examples of pure mathematics becoming “useful”?
- Uses of the incidence matrix of a graph
- Abstract algebra book with real life applications

- Proof writing: how to write a clear induction proof?
- Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals.
- List of interesting math podcasts?
- Important applications of the Uniform Boundedness Principle
- Big List of Fun Math Books
- What does the continuum hypothesis imply?
- Best Fake Proofs? (A M.SE April Fools Day collection)
- Free online mathematical software
- Studying for the Putnam Exam
- Known exact values of the $\operatorname{Li}_3$ function

See “Topological Insulators”, an invention that takes

electronics to a new phase.

From wikipatents:

patents.com link

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

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

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).

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.

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.. đź™‚

(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.

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!

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.

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

AbstractIn 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

â€śbackward dynamicsâ€ť. 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).

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

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.

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.

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.

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.

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.

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.

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.

**1).** Suppose you want to build a $2$-sphere (up to homeomorphism) using convex polygons cut out of paper. It is natural to want to glue two polygons only along their edges and the edges should be glued in pairs. Further, if two edges of two polygons are to be glued, they should be of the same length so we can overlap them. Knowing that the **Euler characteristic** of the $2$-sphere is $2$ tells you that if $f$ is the number of polygons, and $e$ is the total number of edges in all the polygons, and $v$ is the number of vertices in the final piece, then we have $f-e/2+v=2$. (Note that I do not mean $f-e+v=2$.) This tells you something about the resources you should have with you. In fact, a reasoning along these lines shows that the graphs $K_5$ and $K_{3, 3}$ are not planar.

**2).** The Odd Number Theorem. I am not trained in general relativity. But I’ll hazard that this theorem implies that when observing a star we may see multiple images of it due to gravitational lensing. The theorem says that one will always observe an odd number of images. So if you are an astronomer who found an even number of images in an observation, then either there is a flaw in the experimental setup or a hole in the laws of physics as we know it.

Topology plays a huge role in contemporary condensed matter physics, and many top-notch researchers have made a career of it—e.g. Xiao-Liang Qi at Stanford.

There are more applications of topology to condensed matter physics than I can begin to enumerate (the above linked page names several), but I’ll highlight two:

- Dislocations in crystals. These are an example of a “topological defect”—a flaw in a material which cannot be removed by continuous deformation. Such defects are classified by homotopy groups.
- Anyons and topological quantum computing. Alexa Kitaev has proposed a very robust model for a quantum computer in which the stability of the computer (stability being the main challenge in building a practical quantum computer) is protected by topology (similar to the stability of dislocations in crystals).

A good reference for topology in physics is Mermin’s paper and chapter 9 of Sethna’s book.

- Concavity of a function
- Learning Complex Geometry – Textbook Recommendation Request
- Show that $X \times Y \rightarrow Y$ is a closed map. Give an example where projection to the first factor is not a closed map.
- Rotating one 3d-vector to another
- Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$
- Properties and notation of third-order (and higher) partial-derivatives
- Sums of roots of unity
- Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$
- Index notation for tensors: is the spacing important?
- Prove that $\pi>3$ using geometry
- Measure theoretic entropy of General Tent maps
- Translate an english sentence to first order logic
- If every ideal is principal show the same is true for the image of a homomorphism.
- Flipping heads 10 times in a row
- Name for kind of big O notation with leading coefficient