Applications of mathematics All of us have probably been exposed to questions such as: "What are the applications of group theory...".
This is not the subject of this MO question.

Here is a little newspaper article that I found inspiring:


Madam, – In response to Marc Morgan’s question, “Does mathematics have any practical value?” (November 8th), I wish to respond as follows.
Apart from its direct applications to electrical circuits and machinery, electronics (including circuit design and computer hardware), computer software (including cryptography for internet transaction security, business software, anti-virus software and games), telephones, mobile phones, fax machines, radio and television broadcasting systems, antenna design, computer game consoles, hand-held devices such as iPods, architecture and construction, automobile design and fabrication, space travel, GPS systems, radar, X-ray machines, medical scanners, particle research, meteorology, satellites, all of physics and much of chemistry, the answer is probably “No”. – Yours, etc,
PAUL DUNNE,
The Irish Times - Wednesday, November 10, 2010


The above article article seems to provide an ideal source of solutions to a perennial problem: How to tell something interesting about math to non-mathematicians, without losing your audience?
However, I am embarrassed to admit that I have no idea what kind of math gets used for antenna designs, computer game consoles, GPS systems, etc.
I would like to have a list applications of math,
from the point of view of the applications.
To make sure that each answer is sufficiently structured and developed, I shall impose some restrictions on their format. Each answer should contain the following three parts, roughly of the same size:
• Start by the description of a practical problem that any layman can understand.
• Then I would appreciate to have an explanation of why it is difficult to solve without mathematical tools.
• Finally, there should be a little explanation of the kind of math that gets used in the solution.
♦♦  My ultimate goal is to have a nice collection of examples of applications of mathematics, for the purpose of casual discussions with non-mathematicians. ♦♦
As usual with community-wiki questions: one answer per post.
 A: (Dredged up from the murky past...)
Designing control systems usually involves building a logic circuit that has several inputs and one or two outputs.  Sometimes states are involved (sequencing of traffic lights, coin collectors for vending machines), sometimes not.  In designing such control logic, many equations get written down which represent things like "If these three switches are off and these others are on, flip this switch over here".
Once one has the equations written down, (often as a Boolean function,  a map from
{0,1}^n to {0,1}) one has to build the circuit implementing these equations.  Often times, the medium for implementation is a gate array, which may be a field of NAND logic gates that can be wired together, or a programmable logic device, which is like two or more gate arrays, some with ANDS, some with ORS, some NOT gates, flip-flops which are like little memory stores, and so on.
The major question is: are there enough gates on the device to build all the logic represented by the equations?  To this end, computer programs called logic minimizers are used.  They have certain definite rules (related to manipulation terms in Boolean logic) and certain heuristics (guidelines and methods for following the guidelines) to follow in order to minimize the number of, say, AND and OR gates used in representing the equations.
The mathematics of representing any Boolean function as a series of AND and OR gates, and finding equivalent representations, has been developed and used since George Boole set down the algebraic form of what is now called Boolean Logic.  Computer Science, abstract algebra, clone theory, all have played and continue to play an essential role in solving
instances of this kind of problem.  The fact that it is not completely solved is related to one of the Millenium Prize problems (P-NP) .
Gerhard "Ask Me About PLD Chips" Paseman, 2011.02.24
A: Sending a man to the Moon (and back).
Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from Hilbert-Courant by Constance Reid, Springer, 1986, p. 92). 
The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was Richard Arenstorf's  discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE  (see also my answer to the previous MO question).  
Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew. 
A: The error-correction required for cell phones and 3G and 4G devices to work is mathematics!
A: One typical way that GPS is invoked as an application of mathematics is through the use of general relativity. Most people have a rough idea of what the GPS system does: there are some (27) satellites flying in the sky, and a GPS device on the surface of the earth determines its position by radio communication with the satellites. It is also pretty clear that this is a hard problem to solve, with or without mathematics. The basic idea is that if your GPS device measures its distance between 3 different satellites, then it knows that it lies on three level sets which must intersect at a point. This is the standard idea of triangulation. Of course measuring distance is hard to do, and relativity comes into play in many different, nontrivial ways, but there is one way in particular that is interesting and easy to explain.
If one uses the euclidean metric to determine the distance (so, straight lines) from the GPS to the satellite, then it will be impossible to determine the location on the earth to a high degree of accuracy. So instead the GPS system uses the kerr metric, that is the lorentz metric that models spacetime outside of a spherically symmetric, rotating body. Naturally this metric gives a different, more accurate distance between the observer on earth and the satellite. The thing that is surprising to people is that the switch from  euclidean to kerr is required to get really accurate gps readings. In other words, without relativity you might not be able to use that iphone app to find your car in the grocery store parking lot.
People are often surprised and interested to learn that the differences between relativity and newtonian gravity really are observable. Other standard examples are the precession of the perihelion of mercury (which was a famous unsolved problem before the introduction of GR) and the demonstration that light rays do not travel along straight lines by photographing the sun during an eclipse. This last observation demonstrated, for instance, that the metric on the universe is not the trivial flat one.
A: Here are some examples
to quote my favorite one:
In 1998, mathematics was suddenly in the news. Thomas Hales of the University of Pittsburgh, Pennsylvania, had proved the Kepler conjecture, showing that the way greengrocers stack oranges is the most efficient way to pack spheres. A problem that had been open since 1611 was finally solved! On the television a greengrocer said: “I think that it's a waste of time and taxpayers' money.” I have been mentally arguing with that greengrocer ever since: today the mathematics of sphere packing enables modern communication, being at the heart of the study of channel coding and error-correction codes.
In 1611, Johannes Kepler suggested that the greengrocer's stacking was the most efficient, but he was not able to give a proof. It turned out to be a very difficult problem. Even the simpler question of the best way to pack circles was only proved in 1940 by László Fejes Tóth. Also in the seventeenth century, Isaac Newton and David Gregory argued over the kissing problem: how many spheres can touch a given sphere with no overlaps? In two dimensions it is easy to prove that the answer is 6. Newton thought that 12 was the maximum in 3 dimensions. It is, but only in 1953 did Kurt Schütte and Bartel van der Waerden give a proof.
The kissing number in 4 dimensions was proved to be 24 by Oleg Musin in 2003. In 5 dimensions we can say only that it lies between 40 and 44. Yet we do know that the answer in 8 dimensions is 240, proved back in 1979 by Andrew Odlyzko of the University of Minnesota, Minneapolis. The same paper had an even stranger result: the answer in 24 dimensions is 196,560. These proofs are simpler than the result for three dimensions, and relate to two incredibly dense packings of spheres, called the E8 lattice in 8-dimensions and the Leech lattice in 24 dimensions.
This is all quite magical, but is it useful? In the 1960s an engineer called Gordon Lang believed so. Lang was designing the systems for modems and was busy harvesting all the mathematics he could find.
He needed to send a signal over a noisy channel, such as a phone line. The natural way is to choose a collection of tones for signals. But the sound received may not be the same as the one sent. To solve this, he described the sounds by a list of numbers. It was then simple to find which of the signals that might have been sent was closest to the signal received. The signals can then be considered as spheres, with wiggle room for noise. To maximize the information that can be sent, these 'spheres' must be packed as tightly as possible.
In the 1970s, Lang developed a modem with 8-dimensional signals, using E8 packing. This helped to open up the Internet, as data could be sent over the phone, instead of relying on specifically designed cables. Not everyone was thrilled. Donald Coxeter, who had helped Lang understand the mathematics, said he was “appalled that his beautiful theories had been sullied in this way”.
A: A particularly striking application to physics and chemistry is explained in Singer's book Linearity, symmetry, and prediction in the hydrogen atom. The practical problem, in the large, is easy to state: what is the stuff around us made of, and why does it react with other stuff the way it does? More precisely, what explains the structure of the periodic table? 
There is no a priori reason that the elements ought to naturally arrange themselves in rows of size $2, 8, 8, 18, 18, ...$ with repeating chemical properties. This periodic structure profoundly shapes the nature of the world around us and so ought to be well worth trying to understand on a deeper level. 
Physically, the answer has to do with the way that electrons arrange themselves around a nucleus, one of the classic examples of the breakdown of classical mechanics. The Bohr model posits that electrons are arranged in discrete orbitals $n = 1, 2, 3, ... $ with energy levels proportional to $- \frac{1}{n^2}$ such that the $n^{th}$ energy level admits at most $2n^2$ electrons. This behavior $- \frac{1}{n^2}$ can be empirically deduced by an examination of atomic spectra but the Bohr model still does not provide a conceptual explanation of it. 
That explanation comes from full-blown quantum mechanics, which already requires a fair amount of nontrivial mathematics. For our purposes quantum mechanics will be described by a Hilbert space $K = L^2(X)$ where $X$ is the classical phase space (e.g. $\mathbb{R}^3$) and a self-adjoint operator $H : K \to K$, the Hamiltonian, which will describe the evolution of states via the Schrödinger equation.
The simplest case is that of an electron orbiting a single proton, in which case one can explicitly write down the potential. In this case the Schrödinger equation can be solved fairly explicitly and the answer tells you what electron orbitals look like, but it turns out that one can do much better: it is possible to predict the solutions and their properties using representation theory. 
To start with, the Coulomb potential has a spherical symmetry, so this endows $K$ with the structure of a unitary representation of $\text{SO}(3)$. By identifying two wave functions together if they lie in the same representation we can hope to have a physical classification of the possible states of an electron; the idea is that physical quantities we care about should be invariant under physical symmetries (e.g. mass, energy, charge). The action of $\text{SO}(3)$ breaks up the space of possible states based on their angular momentum (Noether's theorem). The corresponding representations have dimensions $1, 3, 5, 7, ...$ and indeed we find that we can decompose the number of elements in each row of the periodic table as
$$2 = 1 + 1$$
$$8 = 1 + 1 + 3 + 3$$
$$18 = 1 + 1 + 3 + 3 + 5 + 5$$
corresponding to the possible angular momentum values allowed at each energy level. Of course these symmetry considerations apply to every spherically symmetric system so the $\text{SO}(3)$ symmetry cannot tell us anything more specific. 
But it turns out there is even more symmetry to exploit. First of all, remarkably enough the $\text{SO}(3)$ symmetry extends to an $\text{SO}(4)$ symmetry. (I do not really know a conceptual explanation of this, unfortunately; I have a half-baked one which I'm not sure is valid.) The irreducible representations of $\text{SO}(4)$ occurring here are precisely the ones of dimensions $1, 1 + 3, 1 + 3 + 5, ...$ and they break up into irreducible $\text{SO}(3)$ representations in exactly the right way to account for the above pattern up to a factor of $2$. Second of all, the factor of $2$ is accounted for by an additional action of $\text{SU}(2)$ coming from electron spin (the thing that makes MRI machines work). 
So representation theory provides a strikingly elegant answer to the question of how the periodic table is arranged (if one accepts that a single proton is a good approximation to a general atomic nucleus). Of course there is much more to say here about the relation between representation theory and physics and chemistry, but I am not the one to ask... 
A: computerized tomography
In computerized tomograhy one meaures X-ray images of a body from different angles. Each X-ray image roughly corrsponds to a projection of the density distribution along a certain direction. To obtain the full density distribution inside the body one has to invert the Radon transform. This is an interesting problem from integral geometry which is also challenging concerning the numerical implementation, since the inverse is known the be discontinuous and hence, regularization techniques has to be employed.
Another interesting aspect of this story is that the mathematical problem of the inversion of the Radon transform was done aound 1917 (by Johan Radon itself) while this was totally unknown to the inventors of computerized tomography as it is used today.
A: cryptography
example: Information send over the internet needs to be secure such that only the sender and the recipient can understand and use it. Example: Man in the middle attack on a bank transaction: You send an order to your bank to pay 100 Dollars to Mr. X. I intercept this transmission and change the order to your bank to make them send me 100 000 Dollars instead. Since every information send over the internet passes through a lot of different computers (gateways), all I need to intercept your message is access to one of those computers. Thousands of network administrators do have such an access (this is grossly simplified of course).
In order to secure the information, me and my bank need to know an algorithm for cryptography. Commonly used are algorithms using public/private key pairs. These consist of functions such that


*

*the bank publishes a public key k,

*I can apply a function f mapping the information inf I would like to send, using the public key k, to an encrypted message f(inf, k). 

*The whole punchline is that the inverse function can only be computed by knowing the private key, which only my bank knows. So only my bank can compute the information inf knowing f(inf, k).
Commonly used algorithms are based on the assumption that there is no efficient algorithm to factorize large numbers, i.e. compute the prime factors of a given large number. The validity has not been proven. So you can 
a) get famous by proving this assumption,
b) get either famous (and provoke a collapse of internet banking) or insanely rich by finding an algorithm that computes prime factors efficiently,
c) get famous and rich by finding an efficient algorithm for public/private key encryption that is efficient and provable safe.
A: computer game consoles
Many computer games today display some sort of 3D real time graphics. There are two important aspects:
a) the display of a 2D projection of a 3D object model needs involved algorithms that calculate what is visible from the viewpoint of the observer, how objects look like from that perspective and calculate shading and light effects on colors. These algorithms need concepts from 3D geometry (linear algebra, vectors, areas, projection operators etc.). Many computer science departments have classes for the involved mathematics.
b) the animation effects of many games are calculated by numerical solutions of partial differential equations describing the physical motion of solid bodies and fluids. In order to animate fluids like water, for example, computer games use finite element approximations to the Navier-Stokes equations. Needless to say, this is a very active area of current research. 
(Computer consoles are actually used in research involving computational fluid dynamics because they are cheap, easy to program and very powerful.)
The last part is also true for automobile design and fabrication:
Car companies need to test new car designs for example for mechanical problems: Are there any parts that will make noise once you drive faster than 50 km/h? This is tested with software that simulates the mechanical parts of the proposed car design using finite element approximations to equations of solid state mechanics. The same technique is also used to simulate crash tests.
The design of the car body is done via CAD (computer aided design) software. This software uses approximation and interpolation algorithms to calculate external surfaces that are as smooth as possible while satisfying boundary conditions that are specified by the designer. These approximations are done e.g. by spline interpolation. 
Numerical approximations of computational fluid dynamics are also used to simulate tests in the wind tunnel. This actually saves a lot of money. (It is also the reason why modern cars all kinda look alike.)
A: circuit design and computer hardware
Example: a robotic arm has to create a complicated electronic circuit by putting a conducting material on a non-conducting base. The robotic arm has to traverse the whole graph that makes up the circuit at least once. In order to reduce the time the roboter needs to create one electronic circuit, the way it has to traverse needs to be minimized, that is one needs a good approximate solution to the traveling salesman problem. I know of examples where improved heuristic approximation algorithms have increased the output by several percents (the student doing the math thesis on this was rewarded by the company producing these circuits with several hundred thousands of dollars. No, it wasn't me.).
A: Gábor Domokos set out to answer the mathematical question: "what are the most probable shapes you will find if you randomly fracture something into pieces (under some reasonable assumptions, e.g. the pieces are all convex, no indentations, etc)?" With Douglas Jerolmack he started working on incorporating the answer from mathematics with actual examples from rock formations and the mosaic of the Earth's tectonic plates. If I understood correctly, studying these examples they understood that the answer would be very different depending on the probability distribution underlying "random fractures". This in turn shed light on the actual geological process of rock formation and could be helpful in answering a major open question in geophysics: "How did Earth’s tectonic plates form?"
I learned about it from this very well-written article on Quanta Magazine.
