Justifying/Explaining math research in a public address I have been chosen by my university to give a 1 hour public research lecture.  Every year a researcher is chosen for this honour.  Traditionally people explain their own research about designing improved airplanes or the life of a peasant in the middle ages or whatever.  I don't feel that I can explain my research in an hour to the general public in any sort of intellectually honest way.
I have decided that it would be more interesting and more useful for me to try to say some things about math research in general.  I hope to explain what math research is like, why it is important to society and how it differs from research in other scientific fields.
I think 3 or 4 really good examples of advances arising from recent mathematical research would go a long way to making my points in an interesting manner.  I am thinking of examples like the page rank algorithm used by Google. Or perhaps modern cryptography techniques such as the RSA algorithm.
Question:  What are some other examples that help show the public why math research is important?
I think it is
important that these be relatively recent examples and that they relate directly to things people experience themselves.
I would be grateful for any suggestions which might improve my talk.
 A: I would not abandon the idea of giving a talk about your own research or the circle of ideas that motivate it.  Almost everything in pure mathematics is ultimately motivated by questions that a broad audience might appreciate.  What are the patterns in the distribution of primes?  Is every curve an intersection of two surfaces?  Which numbers are sums of two squares?  How many fundamentally different ways are there to continuously attach a line to each point on a circle?  What is the smallest area in the plane inside of which it's possible to continuously turn a unit line segment around by 180 degrees?  Almost surely your research arises from some such natural question and its natural generalizations.   I'd try to think of a good, simple, motivating problem, explain why it's a natural thing to wonder about, and where one might get stuck, and what kinds of new ideas this requires, and how one is naturally led to generalize, etc. 
You've got the opportunity to help people appreciate, in some small way, what you actually do.  Why waste it?
A: You mentioned you might talk about how math research differs from science, but perhaps you want to think about how they are alike.  This might make it easier for your audience to relate.  When you get down it, pure scientific research is about describing the structure and behavior of the system in question.  Isn't pure math like this?  I think it doesn't occur to many people  that structure is arguably the key issue in both math and science, and the practical discoveries come after structure is elucidated.  So doing what seems to be "useless" pure math to a layperson is in fact essential, because the pure math describes the structure from which applications spring.
A: I find Catastrophe theory to be a source of compelling examples of the type you seem to be looking for. The applications aren't technological toys (at least, as far as I know)- rather, they're an understanding of how and why some systems can change suddenly.
Build a catastrophe machine- Zeeman's is wonderful, Poston's paper has other ideas, and further ideas are mentioned here:

Here's an interesting application of a cubic: put a bar of soft iron in a mild magnetic field. A slight magnetism is induced in the iron. As you increase the strength of the magnetic field slowly, the magnetism of the iron will increase slowly, but then suddenly jump up after which, as you still increase the strength of the magnetic field, it increases slowly again. If you now DECREASE the strength of the magnetic field, the magnetization will, of course, decrease slowly- PAST the strength at which is had suddenly jumped up. It will then suddenly jump down but at a point where the magnetic field is weaker than when it jumped up. E.e.s call that a "hysteresis loop".
Similar application. You wind the propellor of a toy rubber-band airplane and the rubber-band winds around itself. Suddenly, the entire rubber-band will twist into a spiral. If you now reverse the winding, the force on the rubber-band will go down PAST the point at which it had suddenly kinked before it "unkinks". That is referred to as the "rubber-band catastrophe" (anyone remember "catastrophe theory"?)
Why? Because the equilibrium solutions for magnetic field as a function of induced magnetization and for the force on the propellor as a function of "twist" of the rubber-band is a cubic. Notice the way those functions are going! Induced magnetization is not a FUNCTION of magnetic field (nor is "twist" a function of force) because the cubic would be "lying on its side" and we would have 3 values of induced magnetization for some values of magnetic field. Think of it as $x= y^3- 6y^2+ 9y$. The "switchback" section is between the two extrema for x, 4 and 18. In that region, the "switchback" section that connects the other two is an unstable equilibrium while the other two are stable. As you start increasing the magnetic field, you stay on the lower branch until you are past the local maximum x (in the example above, $x= 18$) and now the value jumps to the other branch. Reducing the magnetic field, you stay on the "upper" stable branch until you hit the local minimum x (in the example above, $x= 4$).

These and similar examples are magical to an audience- you actually see the catastrophe happen, and it's completely mysterious why. Until you explain it mathematically (and the explanation is simple enough for a general audience to understand), and then it becomes obvious.
Different but related magic is the double pendulum. As a young teenager I went to a popular talk on chaos in which this was explained, and I found it to be absolutely magical. Again, the mathematics behind it is accessible, and it's not difficult to convince people, at least implicitly, that this is something which humanity as a whole is richer for understanding.
A: My web page on Popularisation and Teaching has articles and comments discussing a number of issues. Long ago I set about giving public lectures on "How mathematics gets into knots" using knots as a vehicle for explaining to the general public some methods of mathematics. See also the Aims for this Knot Exhibition. Knots were chosen because everyone understands, in some way, what is a knot. 
I agree with trb465 that the notion of structure is a crucial development of the last century, including its investigation using category theory. Often it is thought that maths and science are about numbers, but the notion of structure is much more subtle, and mathematicians have developed language(s) for expressionh, description, investigation, deduction, validation and calculation in this and many areas. Thus it is amazing that the  brilliant images from the Voyager journeys are possible only through the mathematics of error correction, which also applies to CDs and hard disks, and uses complicated algebra to find efficient error correction codes. (You might be able to get images from NASA showing this importance.) 
As suggested above, it could be as well to choose a topic which you love and are familiar with. 
Hope that helps. 
A: you might be interested in Gowers' one-hour talk on the importance of mathematics. there he justifies/explains mathematical research in a very accessible way.
A: It's very valuable to know the material if you want to give a good talk. If your research is pure mathematics, and you decide to give an applied mathematics talk instead, you have a lot of material to learn. Perhaps you could suggest that someone else give the talk? If that isn't an option, you should say more about your background rather than assuming that any applications are going to be equally easy to learn well enough to create your own exposition, and to be able to say that the research itself was exciting instead of just the result.
By the way, there have been many community wiki questions on MO about things like mathematical applications by subject area, math to tell nonmathematicians, etc. Describing your research or field or mathematics itself to nonspecialists, to math majors, or to the general public should be practiced long before you give an invited talk.
A: I‘m all for talking about the applications of math and your research. I think such things can be very interesting. However, let me add one more point. I feel you should also express to the audience why we really study mathematics; that is, it can be a source of immense abstract and intellectual beauty. Of course, some people do like to solve real world problems, but even there the most proximal reward is the joy of seeing an entire solution come together.
I feel that to omit such a point removes the human component of mathematics and places it into the realm of austere practicality. 
I suggest you take your favourite ideas, make them accessible by considerable simplification, and attempt to explain why YOU care and like the mathematics, not why the audience should care.
I have found that most people never fail to respond to the genuine enthusiasm and well-communicated passion of another person.
If practical applications interest you, talk about those. If you‘re much more interested in the wonders and effectiveness of group character theory, speak about that provided you can give at least some illuminating examples. I think this is the only way to be honest.
A: I once attended a talk in which nothing but the classification of Platonic solids was explained to a general audience in order to show what math is like. 
The speaker was doing really great in drawing beautiful pictures at the blackboard. Very little notation was used and no slides where involved.
Even if you didn't manage to follow his arguments to the very end you could read off his fascination of his face and look at the pretty pictures. 
He gave the talk in such a way that not a finished proof was presented but the object where explored together with the audience and steps of the proof where found in a live-discussion. 
Maybe you can tell from what I am writing that this talk was really impressive and reached it's goal to teach that math can be a beautiful and fascinating way of exploring structures (or solving puzzles if you wish to say so).
A: Compressed sensing, see the pediatric MRI example in this Wired piece, http://www.wired.com/magazine/2010/02/ff_algorithm/ (Wayback Machine)
A: There are a number of interesting examples on the NSF Mathematical Sciences Institutes page. 
A: The various answers to my big-list question on Applications of Mathematics might provide inspiration for preparing your talk.
A: David, I'd suggest to use physics or biology as targets.
I mean, try to build a bridge between your research area and the applications... which ultimately could turn into a technological or daily application.
Examples:


*

*Differential geometry -> General relativity (gravitation) -> Fine corrections in GPS devices.

*Lineal Algebra -> Quantum mechanics -> Transistors -> Computer and cell phones

*Chaos theory -> Critical points -> improvement of Weather predictions

*Dynamical systems -> (Population modelling ...)

*Path integrals -> Financial market
And so on...
Good luck with the seminar... and enjoy it!!! 
P.D.: Include graphics, short videos or simulations, cartoons... I'd also suggest you to watch the film Freakonomics, could help you.
A: Perhaps I am projecting (I am about to spend a few hours writing a paper, and hope to make it very accessible while still making it appealing, but am still struggling with the planning stage), but I detect a hint of something which might result in a poor talk.  That something can manifest in various ways, but I will phrase it in terms of goal management.
Some talks suffer from not achieving (for whatever reason) the goal of interesting the audience.  One cause is that the speaker is interested in talking to himself/herself, to reassure themselves that what they are saying is true and interesting to them.  I suspect from your remark on intellectual honesty that you are trying to avoid this or a similar pitfall, that of being so familiar  with your world that you may be a poor guide and even poorer salesman or travel agent to convince others to join your world.
Some talks suffer from not achieving (for whatever reason) the goal of effectively communicating knowledge, or ideas, ore emotions, to the audience.  An obvious trap is attempting to include too much detail, while a less obvious trap is showing excitemen about something while not making it clear to the audience why you are excited AND why they should also be excited.
There are other goals that could be mentioned, as well as techniques to help achieve those goals.  While you do say in your post what you want to do, I have a feeling that you are taking on a little too much by talking about math research in general, and that you will end up with so many goals to achieve that you may be disappointed.  If you talked about math research in a specific area, you might contrast several different modes of research and give an audience member an idea of how they might use one or more of those modes.
For example (and I am being inventive here to make a point) take efforts in number theory.  There are people who will play with symbols on paper to try to find new equalities, inequalities, or other relations between objects.  There are some who will take a general algebraic view and try to cast the problem using different algebraic systems to get ideas.  Some will use analytic methods like calculus to get a handle on how fast functions grow or on how good an estimate of a quantity they can make.  Some might use probabilistic methods to show the existence of a number with certain properties.   Others might employ a geometric  intuition to get a handle on such relations.  Computer programs will be written and run, not to prove things but to provide evidence for or against some conjecture.  Some researchers will comb the literature, trying to find related papers and assemble the pieces like a work of art to create a new result, or clarify an old one.  Others will revisit the literature and provide new proofs in an attempt to improve their own understanding of what they study.  (Note how quickly I generalize to activities that are common to many sciences, and I have not yet mentioned any specific ideas of geometric number theory or algebraic number theory or analytic number theory, yet the different perspectives indicate why there are at least three major branches in that field alone.)
You can talk about all the above, but if the excitement and emotional component of discovery, of repeated trial and failure aand occasional success, if those aspects are missing, much of the audience will  wonder why they are there.  Also, if this is something you are not passionate about, you will have a hard time communicating such passion and emotion to the audience, which I believe is key to a successful talk.  Best to make sure you are very interested in what you are about to say, and not try to force it to fill the air with words.
Find some talks that you believe are good role models and borrow ideas from them; likewise remind yourself of what to do andnot do from talks that are not such good models.  If you worry about the audience understanding, use common analogy honestly  and freely (e.g. "It was like hitting 3 under par!", or "This approach smelled so right, it was like being in Momma's kitchen.").  If you worry about the audience being bored, wake them up occasionally (perhaps with the rare joke, or an Emeril Lagasse-like "Bam! The example demolished that conjecture!", but use sparingly.)
The more I reflect on it, the more I find similarities between your situation and scripting a one hour science documentary.  If you still need advice or suggestions, think about how  the soundtrack of a such a documentary contributes tothe presentation, and what you can use from the approaches they take (repetition, focus, editing, splitting the story into  two paths to create tension, and so on).
Enough blather; hope you find some of it useful.  Good Luck!
Gerhard "Going Back To Goal  Management" Paseman, 2012.08.07
A: From the OP: "I hope to explain what math research is like, why it is important to society and how it differs from research in other scientific fields."
Three goals in one hour are a lot. The first alone is already difficult, but one attempt in this direction is a movie, "Colors of Math". It conveys, to adults, a hint of what it feels like to do math. You could do worse than show it to your audience. Yuri Tschinkel, at NYU, is connected with it and would, I imagine, help you to get a copy. 
A: After a quick glance at your website, I think you should definitely talk about symmetry and invariants and any tie-in to cryptography. 
The general audience loves visual symmetry and you may actually be able to describe some of the higher order symmetries you investigate through basic analogy.
Invariants are also a great topic that I think would be accessible, but that few people might have thought about. In a way, invariants are like the physical laws of conservation (of energy, momentum, etc.). There are some great accessible problems that use invariants (usually of the flavor: notice that in this set-up these two quantities always have to sum to an even number, thus it's impossible for one to be even and the other odd). That type of thinking is so natural to mathematicians, but often very foreign to others. Yet once they see it, they immediately appreciate it.
You could then tie this into cryptography. Surely there are some neat schemes out there that rely on symmetry, seem very convincing, but are broken under some sort of invariant calculation. The talk could progress as: Crypto Scheme, Why it works (via symmetry), The Power of Invariants, How it can be broken. (I've seen very accessible talks in this vein as an intro to basic linear algebra.)
Good luck!
A: Some of the mathematical advances that influenced people's life in XX century are:
Fast Fourier Transform (applications to signal processing, in particular, image processing. Without it
modern computer revolution would be impossible).
Radon transform (which makes computer tomography possible). 
Progress in probability theory (all insurance industry is based on it. And "financial math" as well.
Trading in derivatives is a relatively recent innovation. Based on non-trivial math, stochastic
differential equations).
Progress in control theory (automatic control devices are literally everywhere).
Remarkable progress in coding theory in the end of XX century.
