The correct answer is SUBJUNCTIVE MOOD.
In "All I ask is that you be quiet while I am studying" the action is expressed in the Subjunctive Mood, which is normally used to reflect the speaker's subjectivity. The structure used is the following:
Verb of demand: ask + that + bare infinitive "BE". (Present Subjunctive).
One remarkable characteristic is that in the Present Subjunctive the bare infinitive form of the verb is used with all the persons. For example, "It is paramount that SHE STAY in silence".
