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.