This is both a specific and a general advice thread, since I will likely be able to apply this to other discussions than the one in question. I have a weird situation. I've been asked to give a talk on the research I've been doing as part of a collective celebration of my local university's research. Naturally, and prematurely, I accepted. Some of the professors that I have worked with will be attending, and they are expecting some kind of extravagant bundle of mathematical goodness that they've come to expect from me. HOWEVER. There's a catch. The event is also going to be populated by arts and science and psychology and business and (whatever other areas of study the school supports) students. I want to impress my professors, but I also don't want to alienate/bore/confuse/eviscerate the students of the [strike]lesser[/strike] non-math subjects that will be attending. The question comes down to how to give a rigorous talk (assisted by a slide show thing that I have to prepare) that is still interesting and comprehensible to outsiders. I think this would be a somewhat common debate for teachers. Does anyone have some experience or advice? Any help is greatly appreciated.