I am a math student in the US preparing for an hour long defense of my thesis. I am assuming that the dissertation committee has already read my dissertation by the time of my defense.
- Any advice for preparing and giving the dissertation defense talk?
- What balance between presenting subtle parts of the proofs and a clear big picture of the results obtained should I aim for?
- Would it be more interesting for the committee to hear me explaining more technical parts of the proofs rather then the statements of the main theorems obtained?
I don't think there is a meaningful answer to this question that is helpful across all US math departments and advisors. There is too much variation in what is expected. You need to find out what your committee expects.
That said, if I had to answer without knowing your department or committee, I would recommend trying to focus on the big picture but making sure that you spend most of your time explaining your contribution to that big picture. That's certainly what you'll want to do for job talks, which you're presumably also giving around now.
If the committee is concerned about your understanding of the technical parts of the proofs, they can ask you in the closed exam following the public talk.