I have a question that has been bothering me a bit more over the last few days than ever. What are the core computer science areas? What are the core skills that computer scientists have that researchers in other disciplines don’t have (or computer scientist have an edge over others in them)?
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First Principles 
I don’t think it is possible to answer your question in as definite terms as you seek. This is mainly because the term ‘computer scientist’ begs to be defined precisely; each person has his/her own definition. A discussion of this sort would be akin to a bunch of people discussing about *the* elephant when in fact each of them is seeing a different elephant.
With the above disclaimer in place, let me say that _I_ think that the most important (excuse me for using such a phrase here) skill needed for computer scientists — more than most other disciplines — is a thorough understanding of and the capability to think in and apply /logic/. Please note that I say that this is _needed_, I am not trying to imply that computer scientists actually do have this (or not, for that matter.)
As far as your question about the ‘core’ computer sciences areas is concerned, I think it would be incorrect to classify anything as core or not. Although somethings are clearly more core (sic) than others, I think any attempt to narrow down on the list of core things cannot be completely fair.