I am a Physics undergrad, who is considering the prospects of changing to maths. I was wondering if there is research going on in pure abstract algebra nowadays. I am aware of the fields algebraic number theory, and algebraic geometry which applies algebra, but is there research going on in some specific areas of abstract algebra.
These guys and these guys and these guys seem to think there's still stuff to talk about in algebra. Your university library will probably have the latest editions of them on file. If you want to get a feel for the stuff that's going on right at this second, check those out - even if you don't understand what the articles are about, it's nice to get a glimpse of the buzzwords and general subject matter going on right now. There's also this for reading material, which is even more accessible. You can pick out the algebraic stuff from the list of topics.
To my knowledge the hottest pure algebra topic right now the Langlands program, as mentioned in the comments. Quantum groups and cryptography are applied algebra topics that attract tons of funding (I would take "applied" with a grain of salt here... much of both these topics is as abstract as you would find anywhere else in algebra). There is research happening in just about everything, though.
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Certainly! I hesitate to draw a well-defined line between 'pure algebra' and 'applications of algebra', but I think these fields can be considered 'pure': commutative algebra, noncommutative ring theory, finite group theory, semigroups, Lie algebras, vertex algebras, universal algebra, category theory, and of course representation theory...
To summarize: there is lots of research in algebra going on. Lots.