My web page http://pages.bangor.ac.uk/~mas010/publar.html 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 http://www.popmath.org.co.uk. 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 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.

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.