My background: I got tenure many years ago and have been at a regional “undergraduate oriented” university for over 20 years. I’ve been to scores of interview talks. So, I’ll give some unsolicited advice about these.
Note: what I say doesn’t apply to those who are interviewing for a research oriented job (say, Big Ten or Pac Ten university) nor does it apply for “no research required” colleges and universities.
Our place has a research requirement; you won’t get tenure without a few scholarly articles. But on the other hand; we are realistic; you won’t publish in The Annals of Mathematics if you come here. You’ll spend most of your time with teaching duties (typically 11-12 hours per semester) and some time with committee stuff. So, here are my suggestions:
1. Remember that there might be undergraduate students there. So you might actually DEFINE terms like and “Simple Lie Algebra” at the start, even though there is no need to do so if you are giving a talk to researchers. You might even give an elementary example of such a beast.
2. Even if you think that your talk is “general”, remember that people like me had our graduate courses 20-25 YEARS ago! So, yes, I remember learning about the “Lie Bracket” but I last saw that a long time ago. And since then, I’ve published, but in a fairly narrow area. My “absorb things quickly” part of my brain is flabby, out of shape, and calcified by 20 years of convincing students that .
A few elementary examples might be a good “hook” to draw people into your talk.
The first recommendation is to ASK your contact about the type of talk that they want. Beyond that:
1. First third of the talk; give motivational examples, easy examples; pretend that you are trying to get FIRST YEAR graduate students interested in talking a FIRST COURSE in your area.
2. Second third of the talk: talk to the general mathematicians. We have our Ph. D.s and publications, but many of our research areas are kind of narrow. Our ring theorist might not recall off of the top of his/her head what a “Levi-Civita Connection” is. Our geometric topologist might have forgotten what the Radon–Nikodym theorem says. Our analyst might not remember what a “flat module” is. So “elementary reminders” (the kind you wouldn’t give if you were speaking to a research university faculty department) might be nice.
3. In the last third of the talk, go ahead and speak to the expert/experts in the audience; now is the time to show off. It is ok to lose the rest of us here.
Remember what we are looking for: someone who will explain things well to undergraduates, someone who is in touch with their class (audience in this case) and someone who won’t “die on the vine” due to disinterest in their subject.