What is the ruling on lying to one's mother in order to go to a place to seek knowledge, and upon whom rests the expiation for the oath sworn by the mother that was not fulfilled?

Lying is forbidden, so the liar must repent with remorse, cessation, and a firm resolve not to return to it. Lying is informing about something contrary to what it is, and a person incurs sin if they do it intentionally.

If the lie is accompanied by an oath, it is a yamin ghamus (a perjurious oath that plunges its maker into sin), for which there is no expiation according to the most preponderant opinion, or expiation is required according to some scholars.

As for your mother's oath, its expiation is her responsibility, not yours. You must inform her so she can expiate her oath. If it is difficult to inform her, there is no harm in following those who permit expiation on behalf of another without their permission, so you can pay the expiation on her behalf without her informing her.