One of the questions I got was asking for a clarification of “maximal”, “minimal”, and “intermediate” projections. I’d encourage you to read the summary notes if you haven’t, because I do go into this there, but here’s another version of that, possibly in a bit more detail.
Maximal, minimal, and intermediate projections—so, what the whole syntactic derivation is about is taking a set of lexical items and “arranging” them by combining them together two at a time. So, when you pick up two of these lexical items, you combine them into one, and we need to know what to call the thing we’ve got as a result (the thing made from the two objects we combined). The idea is that each of the lexical items has a bunch of properties, maybe most significantly its category (it’s a noun, or a verb, for example). Once we’ve combined two lexical items into one object, we need to know what the properties of that object are, and it appears that what happens is that the property of the combined object are the same as the properties of one of the things we combined to make it.
So in every combination of two objects to make a combined object, one of the two objects is special, since it’s the property of the special one that determine the properties of the combined object. We say that the features (properties) of the special one “project” up to the combined object—which just means that the combined object has the same properties as the special one had.
So, that’s essentially what it means to say that the features of an object “project” (projéct, as in “form a projection”). The terms maximal, minimal, and intermediate projection just refer to points along the path of a feature’s projection. A minimal projection is the place where the features start, when the features haven’t projected anywhere. This would be the head of a phrase. A maximal projection is the point beyond which a feature no longer projects—so, when you combine two objects and the special one’s features project to the combined object, the *other* object is necessarily a maximal projection because its features didn’t project any higher than that. An intermediate projection is just any point along the path of projection that is neither at the top nor the bottom.
Now, that’s kind of abstract—in terms that are probably more familiar, a “minimal projection” corresponds to the head of a phrase, like the V in a VP. A “maximal projection” corresponds to the whole phrase, the VP in a VP. And an “intermediate projection” corresponds to the nodes in the middle, for example the V′ in a VP. So, the maximal projection of V is the VP it heads, the minimal projection of V is the V itself, and the intermediate projections of V are any V′ nodes between V and VP.