Justifying/Explaining math research in a public address
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.
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.
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