Are PhDs in Physics and Math Limited to Their Specific Research Interests?

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A PhD in physics with a focus on solid-state physics may not have a strong grasp of string theory and other theoretical concepts, suggesting that they would need to learn these topics from the ground up. Similarly, a PhD in mathematics specializing in algebraic topology might lack knowledge in probability and statistics, which is reflected in the tendency of pure mathematicians to avoid engaging with questions in these areas. This raises questions about the breadth of knowledge within specialized fields and the reasons why experts may not address topics outside their primary research interests.
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Hey all

Lets say one has a PhD in physics. This person's research interests include solid-state physics. Is it true then, that this person will be totally clueless in string theory and other theoretical stuff? So he has to learn the material from scratch?

Same let's say with a PhD in math. If this person's research interests is in algebraic topology, then will he totally be clueless about probability and statistics? I have noticed here on the forums, that the pure math individuals do not really answer probability and statistics questions. Why is this?
 
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Perhaps they have better things to do?
 
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