1. They study sea animals in Aantarctica.

[What does 'in Antarctica' modify? 'Sea animals' or 'study'? Are they in Antarctica now? Or though they are not in Antarctica, are they study sea animals at home or at a workplace.]

2. Look at the boy in the room.
[What about 'in the room'? Does 'in the room' modify 'the boy' or 'look'?]

3. They study math in the room.
['In the room' seems to modify 'study' here, isn't it?]

All these seem to be answering the question ‘where’ and thus would be considered adverbial.

But I’d say, in 1, the phrase modified ‘animals’ since it’s immediately following that word. In 2, the prep phrase modifies ‘the boy’ for the same reason.

In 3, however, the prep phrase is clearly telling where they study, and is therefore adverbial, modifying ‘study.’

1. In the sentence "They study sea animals in Antarctica," the phrase "in Antarctica" modifies the location of where they study. It indicates that the studying of sea animals is taking place in Antarctica.

Regarding whether they are currently in Antarctica or not, we cannot determine that solely from this sentence. The sentence only tells us that the study is happening in Antarctica, but it does not provide information about the current location of the individuals carrying out the study. They could either be in Antarctica or studying from somewhere else.

2. In the sentence "Look at the boy in the room," the phrase "in the room" modifies the location where the boy is. It indicates that the boy is located inside the room.

Regarding whether "in the room" modifies "the boy" or "look," it specifically modifies "the boy." It provides additional information about the location of the boy, specifying that he is inside the room. The phrase does not modify "look" in this context.

3. In the sentence "They study math in the room," the phrase "in the room" does modify "study." It specifies the location where the study of math takes place, clarifying that it occurs inside the room.

So yes, in this sentence, "in the room" modifies "study" by providing additional information about the specific location of where the math is being studied.