I wouldn't make the statement Max did -- "Your "after" is a preposition because it's
followed by a noun phrase. Period." But I do agree with his statement that in the case in point, the "after" functions as a preposition. (What comes into my mind is a student's sentence from long ago -- "No one can train a horse better than me." Although this involves "than" rather than "after," I don't think she meant what she said, and I imagined several of the young men in the class wanting to try to train her. Thus, I hesitate with Max's "period" and think that we have to teach students to look at meaning and context.)
This discussion reminds me of several examples that I'm beginning to collect and starting to call "sliding constructions." Consider, for example, the following sentences:

But the most vivid impression left on me this summer by this theater came not from the stage; instead, it came from the rooms underneath the theatre. In this world underneath existed an atmosphere of mystery which made me feel as if I was exploring an old dungeon in a decaying castle.

What is "underneath" in the second sentence? I let students explain it in either of two ways, but before I say what they are, I'm waiting to see what responses I get.

Ed V.