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I'm in the final year of my PhD program (in Applied Math, though this isn't too relevant). I made the personal decision a while back that I would only consider tenure track jobs if they happened to be in my home state. My research is sufficiently applicable that I have industry backup plans that I would be perfectly happy with, so I don't see this as a bad career choice.

Well, lo and behold a tenure track job is available in the math department at my undergraduate school. I'm excited to apply, but I'm wondering if anyone has any specific advice on how to structure my application since I know most of the professors in the department? Is there anything I should emphasize or avoid? Obviously I need to demonstrate that I've "grown up" since I left and could return as a colleague, but I'm not sure how best to do this.

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    Prepare your application as though you were applying to a department with the same characteristics, but where you have never studied. Keep in mind that if they hire you and you go on to bring glory to the department, they will be ten times as proud of you as an alum of their department. – aparente001 Sep 17 '16 at 21:16
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I've seen job talks of people applying for tenure-track jobs at their alma mater. These talks tend to be more tailored more closely to the research within the department.

For example, these applicants tend (more so than others) to make references to specific projects conducted by professors within the department. I've also seen applicants make light-hearted jokes about the department's culture within a job talk before.

They also tend to be more savvy at answering soft-questions about why the department appeals to them and how they would improve it.

Showing that you're an 'insider' or 'one of them' is a useful strategy that I've seen.

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