I want to describe my current research-assistant position saying that "I work under the advisory of John Green". Is this the correct way of saying that my advisor is John Green? If not, what is the better way to state the whole sentence?
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Advisory does appear as a noun in OED, but it is not used to mean "advice" or "supervision", which is the meaning required here. In addition to unrelated uses, an advisory is a body headed by an Adviser (rather like a Directorate is headed by a Director), or which is formed to advise a company on relevant matters.
OED has it as a North American word.
Google's results for "under the advisory of" could include this use: He worked under the Advisory of Defense, for example.
In the OP's example, naming an adviser, one might say
I believe advisement is the word you're looking for. Try this:
"I work under the advisement of John Green."
You also could say: "I work under the supervision of John Green."
I'd also consider using the present progressive tense instead since you're work is ongoing.
"I am working under the advisement/supervision of John Green."
That's how we'd phrase it in the US.