## Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g. math.AT, math.QA, math.CO, etc.

This is a community-wiki question, so please edit and improve pre-existing answers: let's keep it to a single answer for each subject area.

(This is inspired by Terry Tao's recent post about a periodic table of the elements listing commercial applications. He suggested it might be fun to have such a summary for either the MSC top-level subjects or the arxiv subjects.)

I'd like to propose that for areas in which the applications are either numerous, non-obvious, or generally worthy of discussion, someone volunteers to open up a new question specifically about that subject area, and takes care of providing a summary here of the best answers produced there.

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If you're a new user and don't have enough reputation to edit a pre-existing answer, please don't hesitate to create a new answer. Someone will merge for you. – Scott Morrison Oct 26 2009 at 4:50
Okay, answers are rolling in! Let's try to add links where possible to external references. If you can make the first line a stand-alone summary, that's awesome. – Scott Morrison Oct 26 2009 at 5:11
What does it mean to vote up a subject area? I assume you want people to vote up a post if they think one of the applications is really great. – Anton Geraschenko Oct 26 2009 at 15:06
Up vote answers that you think are particularly interesting or surprising. – Scott Morrison Oct 27 2009 at 17:01
I don't see anything on financial mathematics. Maybe it's not an application to the real world? – Bruce Westbury Feb 15 2010 at 21:24

$\mathbf{math.NA \hspace{0.10in} Numerical Analysis}, \mathbf{math.OC \hspace{0.10in} Optimization and Control}$,

$\mathbf{math.PR \hspace{0.10in} Probability Theory}, \mathbf{math.ST \hspace{0.10in} Statistics}$

From Wikipedia: Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience. In many countries, actuaries must demonstrate their competence by passing a series of rigorous professional examinations.

Actuarial science includes a number of interrelating subjects, including probability, mathematics, statistics, finance, economics, financial economics, and computer programming. Historically, actuarial science used deterministic models in the construction of tables and premiums. The science has gone through revolutionary changes during the last 30 years due to the proliferation of high speed computers and the union of stochastic actuarial models with modern financial theory.

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math.GT Geometric Topology: used for topological quantum field theory (TQFT). There is information on applications of this subject area: here.

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math.KT K-Theory and Homology

Used for the purposes (edit: image processing and computational dynamics) in this book and in the Chomp project.

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Math.DS: Dynamical systems

• Modeling flow of liquids, like the animated flow of lava in the movie "Volcano"
• Solving classical few-body problems, though it doesn't help with the modern notorious "two-body problem"
• Heteroclinic trajectories are used in space mission design
• Heteroclinic tangles are used by chemical engineers to get well-mixed reactants
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Another great example of modeling flow of liquids in movies is "Finding Nemo" where actual mathematicians were hired to generate various twirls and vorticities up to high degree of precision. – Somnath Basu Feb 10 2010 at 16:11
Shouldn't there be some mention of applications of stability theory and bifurcation theory to ecology (the predator-prey model being the most basic example)? – Paul Siegel Apr 19 2010 at 19:20
Predicting ecological and climate catastrophes can be done with bifurcation theory (cf www.nature.com/uidfinder/10.1038/nature08227 ) – rcompton Jan 18 2012 at 23:47
Isn't the vortex of the sinking ship in "Titanic" also an example of the first point? [sorry for the spoiler :-P ] – Julien Puydt Jun 29 at 16:29

Math.HO: History and overview: used for

• understanding mathematics as a social and human endeavor

• considering alternative approaches that historically have been used to explore quantifiable relationships

• to teach us that no mathematical ideas appear in empty social space: every idea its a child of its time. It is application of HO to mathematics itself

One real-world application of mathematics is set forth in Bill Thurston's far-sighted essay On Proof and Progress in Mathematics (arXiv:math/9404236v1 [math.HO]), that purpose being, to provide foundations for social enterprise.

With more than 300 references, On Proof and Progress in Mathematics is among the most-cited of all the arxiv's [math.HO] articles.

Nowadays many influential essays in systems engineering (for example) draw implicitly upon Thurston's influential ideas regarding the central role of mathematics in social enterprises ... it is not only mathematicians who are reading and reflecting upon these ideas.

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math.OA Operator Algebras

• (In "Nature") Operator algebras, and, more broadly, operator theory, appear in mathematical models for quantum phenomena.
• (In Engineering) Completely positive maps are used in quantum information theory. There are also many connections between operator algebras and wavelets, which is useful in electrical engineering.
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math.SG Symplectic Geometry

Symplectic integrators are used for numerical simulation of Hamiltonian mechanics. Prominent applications include molecular dynamics, solar system dynamics, computer animation, and a wide variety of problems in mechanical engineering.

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math.FA Functional Analysis

• Used in signal processing for modeling and design.
• Used in machine learning in the design of classifiers (e.g. spam filters)
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math.NA Numerical analysis

• Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements required.

• Numerical analysis is what makes calculators work. (And so much more!)

• We use numerical linear algebra to approximate solutions to discretized versions of complicated PDEs.

• At the heart of Google's Pagerank algorithm is a relatively simple numerical eigenvector computation called the Power Method. The study of large complicated networks (e.g. Facebook) is done using tools from graph theory which again comes back to using the tools of numerical linear algebra.

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math.GM General Mathematics

• The cosmic distance ladder is largely built using elementary geometry (although for some legs of the ladder, more advanced mathematics, e.g. relativity and probability, play a role).
• Mathematics considered as general is the model of precise reasoning, which is widely ( not so widely as it could be or even should be probably...) used in various branches of everyday living as law, economics, psychology, rhetoric etc. Person who do not know mathematics at all usually have small chances to solve his problems in abstract way, which means that every problem may only certain solution related to defined example, without generalisation.
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math.AC Commutative algebra

• Reed-Solomon codes (a type of error correction codes based on polynomials over finite fields - this is why CDs and DVDs still work even after being scratched!)
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"this is why CDs and DVDs still work even after being scratched" Unfortunately, not always. I have some DVDs which cease to work for no apparent reason, with no visible scratches or damage at all! – Zen Harper Feb 25 2011 at 5:47
I think turbo codes are popular nowadays – unknown (yahoo) Aug 23 2011 at 6:29

math.SP Spectral theory

• Spectroscopy (of course)
• To avoid bridge collapse, knowing where the resonant frequencies are is extremely important. :-)
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It may be worth pointing out that the "bridge collapse by resonance" is at least partly an urban myth. In particular, the famous collapse of the Tacoma narrows bridge was due to a complicated aerodynamics phenomenon, and so the credit for understanding its collapse probably belongs to PDEs. (It has often been cited in textbooks as a resonance phenomenon.) On the other hand, the problems with the Millenium bridge in London were indeed due to resonance, apparently coupled with a positive resonance feedback due to the natural reactions by the pedestrians on the bridge. – ABayer Feb 25 2011 at 15:38

math.AG Algebraic geometry

• Elliptic curve cryptography
• Motion planning: Configuration spaces of robot arms are semi-algebraic sets, and algebraic geometry (especially the Cylindrical Algebraic Decomposition) has been used to understand their geometry and design algorithms.
• Algebraic geometry over finite fields is used to construct error correcting codes.
• Statistical models are often semi-algebraic sets, and algebraic geometry can be used to devise tests for the correctness of the model or to fit parameters.
• Birational geometry can be used in the design of NURBS and CAD tools.
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math.DG Differential geometry

• Lie groups are used in robotics (to find the most efficient way to maneuver a robotic arm, for instance).
• Spherical trigonometry is essential for navigation (a few centuries ago, this was THE application of mathematics to the real world - naval empires were built upon this!)
• Finsler geometry can be used in planning shipping routes when ocean currents and winds (as well as the earth's curvature) need to be taken into account to conserve fuel.
• Differential geometry (Riemann metrics + stress tensors) is used in mechanical engineering to study the properties of large membranes, for example, how one should go about building a large tent. Keyword for literature search "elastic membrane", "continuum mechanics".
• Without taking into account the effects of general relativity on the orbiting satellites that make up the GPS system, the locations reported by GPS receivers would accumulate errors of around 10km each day, rendering the system useless.
• Nonlinear control theory makes heavy use of differential geometry
• Quantum theory of atoms in molecules is an application of Morse theory in quantum chemistry.
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 But surely these applications are almost all "obvious"? It would be more of a challenge to find some part of differential geometry that doesn't have any known application! – Zen Harper Feb 25 2011 at 5:50

math.CA Classical Analysis

• Fourier analysis allows one to precisely divide up the electromagnetic spectrum, leading of course to radio, television, wireless, and so forth.
• The fast Fourier transform (and relatives, such as the fast Wavelet transform) is an essential component of many signal processing algorithms.
• MRI is based on inverting the Radon transform.
• Wavelets are used in signal processing (e.g. image compression, edge detection).

The picture is of a Hilbert spectrometer.

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 And the Radon transform is just representation theory of Lie groups: www-math.mit.edu/~helgason/Radonbook.pdf – darij grinberg Feb 25 2011 at 14:53

math.IT Information theory

• Compression; efficient use of bandwidth
• Error-correcting codes protect against digital data corruption from noise, packet loss, physical damage, etc.
• Used in machine learning
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math.PR Probability

• Stochastic processes such as branching processes and HMMs are used to model speciation and extinction (the Tree of Life), evolution of molecular sequences, cell proliferation, and other things in biology. For example, see `Branching Processes in Biology' by Kimmel and Axelrod.
• Here's a somewhat frivolous one (but one that casinos greatly care about): the number of times one needs to shuffle a deck before it truly randomizes.
• used in the world of finance, economics and gambling on a daily basis.
• Random number generation is a key component of many efficient algorithms, and also plays an important role in cryptography.
• Stochastic calculus is used to price options (Black-Scholes formula) and to hedge against risk. (Of course, it is not always applied wisely...)
• Markov chains are used to find uniformly random objects. This, among other things, makes designing an experiment fairest and crypto-systems based on designs securest.
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 Surely it's all over the place in statistics - randomised sampling and the like? – H A Helfgott Oct 27 2009 at 18:24 Yes, but there is a seperate category for statistics. – Graham Jones Oct 29 2009 at 8:59 I think riffle shuffles are used in home games, not casinos, at least in games like poker. I would not want to say that casinos greatly care about the mathematically approachable riffle shuffles, unless you can say that it was historical concern with the effectiveness of riffle shuffles which led casinos not to use that method, which I doubt. I would drop that item. – Douglas Zare Jan 13 2010 at 14:48 Let's not forget about queueing theory. Concrete settings to which the theory applies are e.g. call centers, hospitals, computer networks. Also, stochastic networks can be used to model and analysis e.g. the spread of a disease or social networks. – Pierre Nyquist Aug 23 2011 at 6:05

Math.AP Analysis of PDE

• Partial differential equations are used a lot for modelling systems in biology and medicine, and help describe e.g. animal coat pattern formation (zebras, leopards...), wound healing, tumor growth, spread of a virus in a population, predator-prey systems in ecology, predicting the variations of concentrations of chemicals (hormones, drugs...) within an organ over time...
• PDEs are used in climate modelling, from atmospheric dynamics to ocean currents.
• Radar imaging is based on solving an inverse problem. The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.
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Inverse problems also occur in medical imaging (MRI, CAT scans, etc.) – Michael Lugo Oct 26 2009 at 15:36
...sorry to make a similar comment again, but PDEs are virtually guaranteed to be useful, almost by definition! – Zen Harper Feb 25 2011 at 5:52

math.OC Optimization and Control

• Optimization is heavily used in medicine for cancer treatment, in a technique of radiation therapy called Intensity Modulated Radiotherapy (IMRT) Take a look at this presentation of one of the main researchers in the field.
• Used everywhere in almost every electro-mechanical system (cars, planes, the power grid)
• Kalman filtering and the like are used to enable (e.g.) radar tracking.
• Optimization (and more generally, operations research) is used for management in logistics and analyses/feasibility studies.
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math.LO Logic

• Lambda calculus, the theoretical basis for functional programming (Lisp in particular), was developed by Alonzo Church in the 1930's as part of his research on recursion and the foundations of mathematics.
• Formal verification is used to verify software and hardware in which failure rates need to be as close to zero as possible (e.g. avionics)
• Finite Model Theory is used to design and improve database query systems.
• logical reasoning is widely and sometime non-trivial used in law application in courts, and in should be in parliaments during setting a law systems
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math.RA Rings and Algebras

• Google's Pagerank algorithm is based, in part, on the singular value decomposition.
• Fourier analysis / transforms and linear algebra is at work in the world millions of times per second (video, audio). In particular, creating or displaying a JPEG image requires the discrete fourier transform.
• Quaternions are used in 3D modeling and animation software to represent rotations in a more robust form than Euler angles (helping to avoid transition issues like gimbal lock).
• Every simulation using Finite Element Method uses algebra in very extensive way.
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I think Fourier analysis / transforms / linear algebra / Finite Element Method don't really fit so well in Rings and Algebras (although, obviously, you can put them there if you really insist) - obviously they use algebraic techniques, but their main core is elsewhere (analysis, numerical analysis, etc.) in my opinion. – Zen Harper Feb 25 2011 at 6:04
Why is SVD in "rings and algebras"? – darij grinberg Feb 25 2011 at 14:52
I don't think finite element analysts (and I've worked with quite a few) care anything at all about either ring theory/modern algebra. Also, while it is technically true that the Fourier transform is a product of representation theory, at the level which it is used in signal processing this perspective is not commonly taken. In fact, JPEG uses the discrete cosine transform to approximate images, not the Fourier transform/convolution theorem, and so it is probably better understood using a functional analysis perspective. – Mikola Jun 3 2011 at 22:24

math.RT Representation Theory

• Much of modern particle physics is related to representations of Lie algebras. For instance, Gell-Mann's "Eightfold Way" comes from the representation theory of SU(3) and its associated algebra.
• Almost every application of theoretic physics in solid state physics extensively uses representation theory in description of periodic and quasi-periodic media as crystals, semiconductors etc. In fact solutions of Schroedinger Equation in such cases, numerical or analytical has to be carried in accordance with representation of crystal/quasi cristal symmetry group.

• There are applications of representation theory to three-dimensional Cryo-Electron Microscopy - there is a recent paper of Hadani and Singer about this in the Annals.
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I would be much more interested in applications of representation theory outside of cognate areas. – JosÃ© Figueroa-O'Farrill Dec 2 2009 at 12:52
You can find representation theory in statistics here: projecteuclid.org/… – Kjetil B Halvorsen Jan 19 at 21:05

math.CV Complex Variables

• Conformal mapping simplifies various problems about heat conduction and fluid flow, such as calculating steady temperatures. Modelling the flow of fluids around solid bodies (for example aircraft wings!) can also be simplified by appropriate conformal mappings.
• Tools from complex analysis (phasors, argument principle, conformal mapping) are widely used in analyzing electrical circuits, and stress and strain analysis in mechanical engineering.
• The uniformization theorem for discrete complex analysis (circle packings) allows for efficient and pleasant visualization of geometric data that can be related to surface triangulations.
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math.ST Statistics

• Accurate polling
• Fraud detection: whether financial (e.g. via Benford's law) or voter fraud.
• Used in the world of finance, economics and gambling on a daily basis.
• Predicting consumer preferences (e.g. Netflix prize)
• Experimental design and hypothesis testing (e.g. testing of medical hypotheses)
• Machine learning
• Quality control
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math.CO Combinatorics

• Combinatorics finds applications in computer science, especially in the run-time analysis of algorithms. It has also in recent years found applications in physics, at least in part via its relationship to quantum theory.
• Combinatorial group testing allows one to quickly isolated defects in a large collection of samples by testing batches of samples at a time.
• Combinatorial designs are routinely used to design experiments in applied statistics and quality control.
• Combinatorial optimization in Logistics and Operations research.
• Graphs are used as models for networks (e.g., internet server connections). Applications include searches for least expensive plane tickets, optimizing garbage pickup routes.
• Graph models are used in machine learning.
• combinatorial properties of permutation group was the fundamental part of Enigma cipher breaking by Marian Rejewski before II-nd WW. It is dated now but it was one of the first application of higher mathematic reasoning to cipher breaking, because before usually linguistic reasoning was used instead.
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 Data compression has played a very important role in communications technology. Audio CD's, DVD's, HDTV, I-pods, and many other examples all use data compression techniques. Combinatorics played an important early role here with the development of the very elegant idea of a Huffman tree (due to David Huffman) and Huffman coding. en.wikipedia.org/wiki/Huffman_tree Joe Malkevitch – Joseph Malkevitch Dec 6 2009 at 1:42 Design theory is used to create efficient error-correcting codes. – Douglas Zare Jan 13 2010 at 14:40

math.NT Number Theory

• From Number Theory comes the ideas and theoretical basis for modern cryptography, used to secure communications everywhere from banking to cellphones.
• A more quirky one is SETI (the primes in binary would be a very clear indication of a signal non-natural origin, and would be a starting point for communication).
• There is a gamma ray telescope design using mod p quadratic residues to construct a mask. Gamma rays cannot be focused, so this design uses a redundant array of detectors separated from the mask to reconstruct directional information.
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Weren't continued fractions first used by Huygens to build some clockwork? (In clockwork, one needs to approximate irrationals by the ratios of the number of teeth on two wheels.) I vaguely remember that a twentieth-century mathematician found this handy during the Cultural Revolution. – H A Helfgott Oct 27 2009 at 18:27
I vaguely recall the Chinese remainder theorem was used by early Soviet mechanical computers (basically by embedding the integers in the profinite integers), but don't recall the details. (I once thought that this theorem was used by ancient China to count soldiers in an army, but it appears that story is apocryphal.) – Terry Tao Oct 27 2009 at 18:47

math.GR Group Theory

• Group theory provides methods for understanding the Rubik's cube, and for generating algorithms for solving the cube remarkably quickly from any state the cube may be in.
• Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy.
• Cryptography - various hard algorithmic problems about groups are used to design crypto-systems.
• Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable.
• Potentially fast matrix multiplication; see this MO question.
• Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim, see this site for description and a graphical explanation. If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.
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The Rubik's cube? Real world, come on! :) – Justin Oct 27 2009 at 17:36
Hey, I solved the Rubik's cube the first time by using group-theory considerations. It took me about 20 minutes to solve a cube. (Then I learned an actual method.) – Harrison Brown Dec 2 2009 at 3:57
I got interested in group theory from an article about Rubik's cube in Scientific American when I was in school. I think I learned the concept of "conjugation" from that article, actually. – JosÃ© Figueroa-O'Farrill Dec 2 2009 at 12:55
Would you happen to remember which Scientific American article? – Paul Siegel Apr 19 2010 at 19:08
See mathoverflow.net/questions/94907/… for applications of group theory to math. biology – Alexander Chervov Jun 29 at 21:15

math.CT Category Theory

• Category Theory helps design modern and novel programming languages, that end up being able to do optimizations based on mathematical theorems, and even allow provably correct code with less effort than using other techniques. For example, the language Haskell makes use of many ideas from category theory.
• There are deep connections between category theory and logic, in the sense of computer science.
• Feynman diagrams form a (monoidal) category.
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math.AT Algebraic Topology

• Algebraic Topology finds applications in sensor network design, coverage analysis for sensor networks, and in expanding data analysis techniques to give better visualizations for large data sets.
• It has also been applied to computer vision and pattern recognition algorithms, for instance here.
• Algebraic Topology can be used in robotics. Motion planning and behavioral algorithms for robotics have been studied with topological tools.
• Knot theory is used when dealing with protein folding and other analysis of DNA function. There are enzymes called 'topoisomerases' that change the knottedness of loops of DNA. In fact, when bacteria (which have circular 'chromosomes' called plasmids) reproduce, they make use of an enzyme whose specific role to to unlink Hopf links! There are antibiotics that target this enzyme.
• Model categories have been used in the study of concurrency. See this paper by Gaucher.
• Nash's proof (Ann. of Math, Vol. 54, No.2; 1951) that every finite non-cooperative game has an equilibrium point in mixed strategies is a direct application of Brouwer's fixed point theorem, and spurred a great deal of interest in applications of game theory to economics (cf. this survey article). Game theory itself has applications in computer science and mathematical finance.
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Here is a good page on this by an expert: math.upenn.edu/~ghrist/index_files/research.htm. Persistent homology in particular is a rapidly developing locus of applied algebraic topology which has attracted the attention of some big names like Shmuel Weinberger and Stephen Smale. – Paul Siegel Apr 19 2010 at 19:32
The "designing robots" feature has been claimed as "motion planning" by the algebraic geometry answer. – Elizabeth S. Q. Goodman Mar 5 2011 at 22:15
There mere fact that algebraic geometry, too, is used to understand configuration spaces in robotics does not mean that algebraic topological techniques are not also in use. – Mikael Vejdemo-Johansson Mar 6 2011 at 16:39
My understanding is that real algebraic geometry is used to study e.g. the positions a robot arm with various joints and segment lengths can take by expressing this as a system of polynomial equations to solve, whereas topologists such as Ghrist have used the notion of configuration space for e.g. studying how several robots could move around a factory floor (perhaps on a set of tracks and/or with obstacles) without bumping into each other. – Patricia Hersh Jun 29 at 11:56

math.MG Metric Geometry

• Discrete sphere packing solutions lead to error-correcting codes.
• The earthmover metric is used in image recognition and classification.
• Triangulation using the Euclidean metric is used for navigation (and nowadays, in GPS systems)

The Banach fixed point theorem for contraction mappings has a beautiful application in image compression, called fractal compression. One starts with a complete metric space $X$ of images with Hausdorff metric. Then for a given image $x \in X$ one finds a contraction mapping $A: X \to X$ with (unique) fixed point $x$. To do this, one considers self-similarities in the picture (that's why it is called fractal compression).

Then we get rid of the original image and store the map $A$ only. To reconstruct the image, one starts with any $x_0 \in X$ (for example an image which is all black or all white), and applies $A$ several times. The result will be close to $x$.

When I (Evgeny Shinder) first learnt this in high school (my friend and I implemented fractal compression as a final project for a programming class), I was fascinated how such abstract math can be applied to such a concrete problem as image compression!

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 The taxicab metric is something many of us think about when driving. :) – Ryan Budney Jan 20 2010 at 23:33