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.
