Why does Gertrude, Hamlet's mother, urge Hamlet to stay home and not return to school?
Gertrude and Claudius make the request together in Act I, scene ii of the play. After discussing Hamlet's grief over the death of his father, Claudius and Gertrude gently scold him for still seeming so mournful and encourage him to "lighten up." After this, Claudius says the following:
For your intent
In going back to school in Wittenberg,
It is most retrograde to our desire:
And we beseech you bend you to remain
Here in the cheer and comfort of our eye,
Our chiefest courtier, cousin, and our son.
The message is that, because of the death of the King, they want Hamlet to stay close and be part of their everyday lives, so that the may comfort each other. This statment of Claudius' is followed by a statement from Gertrude:
Let not thy mother lose her prayers, Hamlet:
I pray thee stay with us; go not to Wittenberg.
Gertrude seems to want Hamlet close, for comfort. Claudius, however - as we will learn later - wants Hamlet close for safety, his own safety.