Open problems in physics today

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We don't understand the Pauli Exclusion Principle at all.

We do not understand metals from first principle. The simple nearly free electron approach works ok, but refinements on it fail. We are unable to calculate band gaps from first principle.
 
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Claude Bile said:
The big cookie in physics is resolving the apparent contradictions implicit in the theories of General Relativity and Quantum Physics.

Claude.

_Apparent_ contradictions?
Are you stating that the problem is merely a paradox?