adjective clause and the object in the sentence

Question

The mosquito moment is the part of the presentation people remember the most in the conference.

people remember the most is an adjective clause, so is placing that or which in front of it optional? And does the PP in the conference go with the adjective clause or say it's part of the adjective clause, is that right?

And what is the object of the verb remember, is it the phrase the mosquito moment (seems a bit far) or the phrase which precedes it the presentation?

I was told that adjective clause modifies the thing that precedes the clause, does that rule always apply?

Answer 1

The mosquito moment is the part of the presentation [people remember the most in the conference].

The relativised element is object of the relative clause so, yes, it is omissible. The antecedent of the omitted relative word is the nominal “part of the presentation". The PP is an adjunct, an omissible element in the relative clause.

The object of “remember” is “part of the presentation”, which is co-referenced to the subject NP “mosquito moment”

Answer 2

Starting with your last questions first:

The object of the verb "remember" is "part", "the part [of the presentation]", and "of the presentation" is a prepositional phrase used to modify "part" (clarifying what kind of part is meant). The second prepositional phrase in the sentence ("in the conference"), could either be considered part of the clause beginning with "people remember....", or it could be understood as a separate, additional modifier of "part". There really isn't much difference in meaning between those two interpretations.

As to your first question, about whether that or which is optional: in your sentence, "that" or "which" can be omitted before the clause "people remember the most". This is common in normal idiomatic English. Also, according to englishgrammar.org:

The relative pronoun that can be left out when it is the object in a relative clause. Here is the example they give,

This is the woman that we were talking about.

OR

This is the woman we were talking about.

Their example is analogous to the sentence about the mosquito moment, so we do not need to use "that" or "which".

It is true that the same website also states, as given in Zeeshan Ali's answer,

That cannot be dropped after nouns. (same source)

Since "woman" is a noun, I am confused by their apparently contradictory rules. I would go with common usage, and say that the sentence you quote is correct.

Answer 3

That as a relative pronoun:

The relative pronoun that can be left out when it is the object in a relative clause.

This is the woman (that) we were talking about

And, therefore, your sentence need not include that/which as follows;

The mosquito moment is the part of the presentation (that) people remember the most in the conference.

The above sentence could have multiple contexts of which here are two;

1. The mosquito moment is the thing (object of remember) which people remember the most.
2. the part of the presentation is the thing (object of remember) which people remember the most.

An adjective clause modifies often the thing that precedes it, but there also the cases like linking verbs and Demonstrative Pronoun when that does not hold.