I would like examples of technological advances that were made possible only by the creation of new mathematics. I'm talking about technology that was desired in some period of history but for which the mathematics of the time were not enough.
It would be great if I could have an example in which the technology needed was associated with an industrial problem, and it would be even better if the technology developed had a significant impact in society.
One of the reasons we don't all speak German today is the combination of elementary group theory, and universal Turing machines.
Per a request below, I'll elaborate: The British, French and Poles had a joint intelligence effort against Germany during the 30s (at that time breaking cyphers was considered a liberal-arts high classes hobby). The French, using traditional cloak and dagger techniques, managed to get their hands on the German cypher machine (Enigma) plans - but did not know what to do with it. The Polish cypher bureau however had an amazing idea - they'll let math students play with it. From a group theoretic point of view, the machine worked by taking products in S_{letters}. However, the machine designers did not notice that the products were all of the form A^{-1} R A - (where A is constantly changing but R is not) which enabled Poles to discover the cycle structure of R. You can look here for more details.
When the Germans invaded Poland, the Poles let the French and the British on the secret, and the British took over the operation. The Poles used machines (bombes) from the very start to break the codes, but as the Germans improved the coding machines, the need arose the reconfigure the breaking machines as they run. Luckily for everyone involved, the head cryptanalysis on the British side was Turing, who built such a machine: Colossus.
Finally, a funny bit which I hear in Bletchley park last year: "everybody" knows that at the end of the war, Menzies and Churchill ordered the Collosi destroyed and the plans burnt, and yet they were rebuilt a few years after they were declassified - how come ? The easy part is that some of the engineers saw it coming and took backup plans home. However, it is much funnier to discover that the Royal post office continued to manufacture the parts until the 70s, and they were still in stock.
One example of mathematical achievement that has application to the "industry" of medicine is the Radon transform, that enables to produce x-ray tomographies.
The Radon transform of a plane section of your body is the set of information that you obtain after shooting x-rays through all the possible lines contained in that plane, and recording the intensity of the ray that comes out at the other side. More formally, its an integral transform: it's the result of integrating the density of the body tissue along each of the lines.
Johann Radon defined his transform in 1917, and calculated a formula for its inverse. That is, he deviced a way to recover from the transformed data the density of tissue in each point of the plane section. The inversion formula involved a lot of calculations that couldn't be handled at that time.
Cormack and Newbold implemented the idea with the aid of computers to handle the calculations. They got a Nobel Price in 1979 because of this.
Public-key cryptography.
Also, compressed sensing.
The work of Oliver Heaviside in electrical engineering is a prime example.
He first of all invented an "Operational Calculus" to solve differential equation, and did non-rigorous stuff that were beyond the pale from the mathematical point of view. But they did work, and Heaviside faced much stiff opposition from the British mathematicians of the time. Later this work was justified formally with the development of Laplace and Fourier transforms.
The Dirac delta function, its derivatives and the paraphernalia were all conceived by Heaviside. The Delta function being for a sudden impulse of current such as in lightning or a spark, etc.. This was again later used by Dirac, and justified later in the theory of distributions.
I can't think of a better example. The problems of electrical distribution, network theory, and above all, all electrical communication systems, arose out of this theory. Even up to digital signal processing.
First it was built non-rigorously by the engineer Heaviside, and it was put on sound footing only much later.
Basic example from operations research: efficient resource allocation is important in any society, so one can argue that civilizations of the past (and the present) have been wanting in part because of the absence of good allocation heuristics. Linear programming is a first major step toward sound quantitative analysis, and the fact that it was known on one side and not the other apparently contributed to an allied advantage during World War 2.
The invention of logarithms in 1544 and later "popularized" in the 1600's, simplified calculations that were practically unfeasible beforehand. The following paragraph is quoted from the Wikipedia entry on logarithms:
Their use contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, they were used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides the utility of the logarithm concept in computation, the natural logarithm presented a solution to the problem of quadrature of a hyperbolic sector at the hand of Gregoire de Saint-Vincent in 1647.
The invention of the Sierpinski carpet by Sierpinski in 1916. Who knew that cell phone antennas would later be based on this shape?
Without the development of error correction codes and data compression codes most of the recent developments in communications technology (cell phones, high definition television, audio CD's, DVD's) would not be possible. Two big pioneers in this were Marcel Jules Edouard Golay and Richard Hamming who developed one of the early practical error correcting codes (especially, Golay's codes were impressive) and David Huffman whose pioneering work helped with the dramatic developments in data compression we see today.
Basic information about Hamming codes can be found here:
https://en.wikipedia.org/wiki/Hamming_code
Basic information about Huffman codes is also available on en.wiki.
Feedback control theory. In the 1920 when people at Bell labs tried to build amplifiers, they oscillated; and oscillators went unstable. Then the likes of Bode and Nyquist came up with the idea of using complex variables to develop stability criteria, and classical control theory was born. With it the use of electronics in telecommunications became possible.
There are many examples from computing but I guess you're after areas that have mathematical interest in their own right.
I think I good one is the Monte Carlo method. The idea has been around since Buffon's idea of estimating $\pi$. But it became important leading up to, and during the Manhattan Project, when Fermi and Ulam both came up with the method as a way to make impossible seeming integrals over high dimensional spaces tractable. For example, those involved in neutron transport. Today it's in use everywhere from finance to 3D graphics (with Pixar holding the patent on its application to ray-tracing).
There have been all kinds of interesting variations such as Markov chain Monte Carlo and various 'stratification' strategies to increase reliability through 'even' sampling, and Las Vegas algorithms that guarantee correct results. There have been some interesting recent 'pure' applications to mathematics, eg. allowing 'exact' samplings of spaces such as the domino tilings of diamonds. And randomised algorithms are routine in areas like number theory - eg. primality testing.
With hindsight it seems like an obvious idea but I think that at the time the idea of using random numbers to compute a non-random quantity was a pretty big shift of mindset.
The Manhattan Project probably "had a significant impact in society".
Information Theory: the original 1948 paper by Shannon is clearly focused in putting the analysis of how to deal with information on sound mathematical basis.
Contemporary work in biology has lots of examples. With the wealth of sequencing data coming out of the machines at ever lower costs there is a huge need for new methods and models to analyze and understand the data.
Google is another good example. Better (or even acceptable) internet search was desired and it took a mathematical advance to move the technology forward.
Ocean navigation is greatly helped by our ability to integrate the secant.
Methods for estimating risk (financial or otherwise) are important for decision-making, i.e., one often wants to anticipate the consequences of ones actions. Some modern statistical techniques have been motivated by actuarial questions in e.g., the insurance industry. Similarly, in the financial sector, people have developed quantitative tools for choosing hedging strategies to manage financial risk. (Recent events suggest that these tools should be used with some caution.)
Extremal problems in ODE's and Pontryagin's maximum (minimum) principle. I think there should be many other examples from Soviet school, where mathematicians were sometimes forced to work on problems of "people's interest" (like optimal control in this case).
I suppose that this is an example that doesn't pack the historical weight of what many others have said, but topology has played an interesting (and up-and-coming) role in computer graphics and network sensing. This site details some of these applications and how they are currently changing various technologies. For example, there are a few examples of persistent homology being using to detect network coverage (i.e. Cell Tower Coverage) a la Ghrist and de Silva (2007) and for feature detection in MRI. Similarly, the ALICE project has used various Laplace-Beltrami operators for Computer Graphics applications as well as the use of Cell Complexes in surface reconstruction. On the other hand, there are examples of homology being used to characterize inviscid flows and in Computational Statistics / Statistical Mechanics (Chapter 5).
Again, these probably don't hold the weight of some of the other answers and at the moment, we don't know how these mathematical innovations will push technology forward; however, they represent the potential future for mathematics as a driving force for technology.
Artillery cannot be developed without proper notion of reference frame, for example Cartesian one, which is needed in order to properly define notion of a curve given by equation. So before Cartesian reference frame was introduced, and notion of relation between equation of notion and drawn curve was introduced, artillery, and any knowledge working with trajectories was blind and unable to form useful domain of engineering beside some pure empirical rules. In fact it remain in this state long ago after even if mathematics was possible to use equation of motion to develop proper description of trajectory.
Statistical process control: developed because of the needs of the factory process to control variation of all the parts that make up the final product. It contributed greatly to the rise the Japanese economy, post WWII.
Hydraulic press, mill, simple mechanics, gearwheel, clocks in order to be made has to be projected, and very important conception here is ability to count mass or volume of parts for this mechanisms. Hellenic mathematician Eudoxus was one of the first who correctly define integral. It allows to precise describe relation between mass and volume. Unfortunately his works was forgotten or misunderstand, and was not intellectually accessible during Middle Ages, so large parts of technology was impossible reconstruct although was present before!
It is example of technology and mathematical knowledge which was invented, and forgotten and then invented again, whit straight ties to technology!
This, and many more examples of such things may be found in excellent book of Luccio Russo "The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn"
One of the most important tools used in Computer Science is the Fast Fourier Transform, which basically made much of image processing possible - described by Gilbert Strang as "the most important numerical algorithm of our lifetime". Note however that unknown to the inventors James Cooley and John Tukey it seems that Gauss actually had already discovered the algorithm (see this paper for more details) but not published it!
Much of modern medical imaging is only possible due to mathematical advances - for a flavour here are a few example projects from the University of Eindhoven:
The specific technologies listed in the Compressed Sensing Hardware page ( http://sites.google.com/site/igorcarron2/compressedsensinghardware ) could not have existed or have a justifiable theoretical background before 2003-2004 the date of the first papers on compressed sensing by Donoho, Romberg, Tao and Candes.
Only time will tell on how much of an impact they will have.
When steam engine was constructed in fact there was no theory of thermodynamics. Heat engines theory was developed by Carnot which was 10 years old when steam engine was constructed by James Watt. So as technology here was present and works well without any mathematics, there was no means of theory for such mechanisms, and then no other than empirical by trial and error way to develop better solutions.
Statistical design of experiments: This was first developed by Fisher in connection with agricultural experiments, later the development continued by George Box and others in connection with industrial experiments. This led to ideas such as factorial designs and fractional factorial designs. Later Taguchi developed what is now called https://en.wikipedia.org/wiki/Taguchi_methods in connection to control and bettering of quality.
Magnetic Resonance Imaging (MRI) associates certain spatial changes in composition of materials with a visible image which may be represented by some real signal (vector) $m\in\mathbb{R}^N$. The vector $m$ often can be represented by a small number of coefficients in some basis $\Psi\in \mathbb{R}^{N\times N}$ (through a wavelet transform for example). However, the MRI signal $m$ is not measured directly but through some number $M$ of approximate samples of the Fourier transform
$$\hat{f}(\xi)=\int f(x)e^{2\pi i x \cdot \xi} dx,$$
these being denoted by $\mathcal{F}_{u}m$, where $\mathcal{F}_{u}:\mathbb{R}^{N}\rightarrow \mathbb{R}^{M}$. In one dimension these Fourier measurements might correspond for example to a sub-sampled DFT matrix with entries powers of $\omega=e^{\frac{2\pi i}{N}}$, so that
$$(\mathcal{F}_{u}m)_k=\sum\limits_{n=0}^{N-1} m_{n}\omega^{kn},\ k\in S\subset [0,N],\ |S|=M.$$
Under appropriate conditions on $\Psi\mathcal{F}_{u}^{*}\mathcal{F}_{u}\Psi^{*}$, with Fourier measurements $b$ observed, and with coefficients $\Psi m$ $s$-sparse ($\Psi m$ has at most $s$ non-zero entries), a primary method of retrieval is by solving the optimization problem $\min\limits_{m} \|\Psi m\|_1$ subject to $\|\mathcal{F}_{u}m-b\|<\epsilon.$
Research in compressed sensing explains under what conditions this recovery is possible, and now accounts for a ten times speed up in imaging times from about ten years ago before systematic studies of this optimization problem.
Markov chains have applications in speech recognition (see A tutorial on hidden Markov models and selected applications in speech recognition by Rabiner), analysis of DNA sequences (see Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids by Durbin, Eddy, Krogh, Mitchison), in Google PageRank engine etc, see Wikipedia article for a longer list of applications.
Human lives were saved on the former Yugoslavia minefields by using software developed for Arithmetic-Geometric Parallel Processor (GAPP). It was using the finite window image covariant transformations.
Here is an ironic twist. The technology progress for ordinary (basically sequential) computers was such, that the said software was transferred onto them (from the parallel processor). This has meant an irrational drawback for the image transformation theory, including the critical mathematical part of it. This way, nobody is interested in attacking the mathematical problems which arguably are the most important in the whole mathematics, as indicated by the promise of the early development.
(Also, the computer technology is a big loser).