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.
 A: math.MP Mathematical Physics: this subject already uses several of the other already mentioned categories, like group theory, functional analysis, applied to physical theories like quantum field theory. But since the intention is to see concrete applications to the real world taken from a mathematically rigorous framework, then we could mention

*

*In the first place Noether's theorem: "every differentiable symmetry of the action of a physical system has a corresponding conservation law".

*Onsager reciprocal relations: They "express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists". Onsager's contribution was to demonstrate that not only is $L_{\alpha \beta}$ positive semi-definite (the Onsanger matrix of phenomenological coefficientes), it is also symmetric, except in cases where time-reversal symmetry is broken.


*

*The issues about proving thermodynamical statements strictly from statistical mechanics. This became one the biggest discussions in mathematical physics on its time, between Ludwig Boltzmann and Ernst Zermelo (see the book of the colleted works of Ernst Zermelo, volume II, edited by Herausgegeben von, Heinz-Dieter Ebbinghaus and Akihiro Kanamori, Springer, 2013).

*The existence of anti-particles by P.A.M. Dirac, which was first a mathematical result. Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of antielectrons, themselves discovered four years latter, by Carl D. Anderson.

*The Universality of the Feigenbaum constants, proven by Landorf ( Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc 6 (3): 427–434.), with some corrections by Eckmann and Wittwer (Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics 46 (3–4): 455).

*The very concept of Universality, together with scaling, introduced by Kadanoff (see for example, Physica A 163 (1990) 1-14 "Scaling and Universality in Statistical Physics"). Roughly speaking, scaling is about the description of changes in the behavior of physical phenomena, in terms of adimensional constants, and how do they scale with them, as is the case of the Reynolds number. Universality concerns the invariance of properties for different dynamical systems, independently of some other physical details. For example, (see the aforementioned paper of Kadanoff), any perturbation which does not drive away a Hamiltonian near a critical point is deemed irrelevant, and all the Hamiltonians with any such kind of perturbations are said to belong to the same Universality class.

*In the speculative region (as of today), the prediction of a possible second Island of Stability, where some new stable chemical elements might be found, with possibly interesting physical properties (see, for example, Zeitschrift für Physik 1969, Volume 228, Issue 5, pp 371-386 "Investigation of the stability of superheavy nuclei around Z=114 and Z=164", by Jens Grumann, Ulrich Mosel, Bernd Fink, Walter Greiner).

*Even more speculative, but still valid mathematical results, some solutions from Einstein's General Relativity equations, allowing for closed timelike curves (i.e. time machines to the past), such as Gödel Spacetime (see, Review of Modern Physics, volume 21, Number 3, July 1949, "An example of a new type of cosmological solutions of Einstein's field equations of gravitation", Kurt Gödel).

*Not yet published as mathematically solved, the strict mathematical derivation of turbulence, from the Navier-Stokes equation, or otherwise (turbulence does not have a complete mathematical theory, to the extent of my knowledge). Navier-Stokes does not even have a theorem for the following problem: "Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations."

*Along similar lines for mathematical completeness, is the Yang-Mills existence and mass gap: "Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$. Its importance comes from the fact that it is the simplest Quantum Field Theory available (as far as I know), and it does not need to assume the existence of quarks.

A: math.KT K-Theory and Homology
Used for the purposes (edit: image processing and computational dynamics) in this book and in the Chomp project.
A: 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!)

A: 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.

A: math.GT Geometric Topology: used for topological quantum field theory (TQFT). There is information on applications of this subject area: here. In chapter 6 of Topological Quantum Computation edited by Zhengan Wang CBMS 112 availabe at http://www.ams.org/bookstore/pspdf/cbms-112-prev.pdf It is argued that TQFT is relevant to the real world due to emergence phenomenon.
Also at this site: http://web.ornl.gov/sci/ortep/topology.html geometric topology is used in crystallography. (Carroll K. Johnson and Michael N. Burnett: Crystallographic Topology - The Topology of Crystallographic Groups and Simple Crystal Structures. Here is a link to a Wayback Machine snapshot.)
A: 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 a description. 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.  

A: 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] 3 is an application of Morse theory in quantum chemistry.

A: 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, pattern recognition algorithms (for instance here), and topological data analysis.

*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.

A: 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.

*The study of the orbits of the permanent and the determinant (thought of as points in the space of polynomials) is the central idea in Valiant's algebraic version of P vs NP, and the representation theory of the relevant coordinate rings of the orbit closures is a leading approach by Mulmuley and Sohoni.  There are many references, but here are two for starters: report, ICM paper
A: 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.

*Finite projective spaces and Galois geometry are used in random network coding and cryptography.

A: 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.

A: math.ST Statistics

*

*almost everywhere

*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
A: 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.

*Projective geometry and implicitization is used in 3D image reconstruction from multiple camera views.

*Geometry and Representation theory of tensors can be used in Physics, Computer Science, Statistics, Phylogenetics, Psycometrics and more. Here is a summary from 2008: report
A: 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.

A: 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.

*With the emergence of SLE (Schramm-Loewner evolution, conformally invariant curves), we can model the critical interfaces of various systems such as Ising and Percolation.

*Evaluation of complicated real integrals that show up in engineering and physics via contour integration.

A: 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.
A: 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.

*Tropical algebra has been used to design product-mix auctions and to calculate demand (see Baldwin-Klemperer).

A: math.PR Probability 


*

*Probability theory is used in information theory, error-correcting codes, compression algorithms, machine learning, probabilistic algorithms

*Physicists use probability in Quantum Mechanics and Statistic Mechanics

*Queuing theory is used to analyze telecommunication networks.

*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.

A: 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.


*Finite elements method (a version of multigrid numerical aanalysis) is all pervasive in construction and achitectural stability analysis, and also in other fields of engineering like engine design).
A: 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

*Pseudo-Anosov braids are used to design efficient methods for stirring viscous liquids (see for example http://arxiv.org/pdf/nlin/0603003.pdf)

A: 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.

A: math.SP Spectral theory


*

*Spectroscopy (of course)

*To avoid bridge collapse, knowing where the resonant frequencies are is extremely important. :-)

*Shape and model recognition

*Network analysis and security (spectral graph theory)

*The design of quantum wave guides

A: 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)

A: 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.

A: 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

A: 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 is 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, 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.
A: 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

A: 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!
A: 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.
A: 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.

