Many (most? all?) North American graduate programs have some form of qualifying exam (which goes by different names at different institutions) whose goal is to establish a baseline knowledge of the kind that you ask about.  Typical topics are algebra and real and complex analysis.  

However, there is no law (natural or human) saying that a person will remember what they have once been taught, and if one's goal (in a colloquium, or equivalently, any lecture to non-sepcialists) is to be understood, it will not be any use to appeal to some alleged common knowledge specified in a document, or a qualifying exam syllabus, or anywhere else.  

I think a more realistic solution is to develop mechanisms for explaining ideas which appeal to a wide range of audience members, and include many different ways for them to try and understand what you are talking about.    This is never easy, unforutnately, but it can become easier with
practice.  I think one thing to remember is that simple geometric ideas are often easier to communicate than algebraic ones, and are (in my experience) more likely to appeal to a wide range of audience members.    Also, I think one should be especially receptive to the idea that audience members will be confused by your notation, and thus one should adopt simple notation and be prepared to explain it.  (So my sympathies are with you in your attempt to divine the meaning of $\hat{f}(t)$!)