Is it permissible to record university lectures for the purpose of note-taking due to the lecturer's fast pace, despite his refusal and the absence of any out-of-syllabus content?

It is not permissible to record someone's speech, whether religious or worldly, without their permission and knowledge. The Prophet, peace and blessings be upon him, said: "If a man speaks to another and then turns his head, it is a trust." This means that the words spoken by your companion are a trust with you, and if you disseminate them without his permission, you have betrayed the trust. Recording without the speaker's permission is deceit and a betrayal of trust, and disseminating it to others is an increased betrayal and a violation of that trust. Therefore, recording a conversation without the speaker's knowledge and permission is an act of wickedness, betrayal, and a blemish on one's integrity.