Is computer algebra or symbolic computation an active area of research? I'm interested in doing  PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to industry after my PhD? I'm also based in the USA.
I apologize if this question is kind of "bare", I'm a  PhD student at a university that does computer algebra, and I just want to know what I'm getting into. Thanks!
 A: Two books and a journal:

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*Invitation to Nonlinear Algebra by Michałek and Sturmfels. Computational nonlinear algebra and a variety of applications (“variety”).


*Applications of Polynomial Systems by Cox et al.


*SIAM J. on Applied Algebra and Geometry
These are recent which may indicate that the subject is active, both as a mathematical subject (eg, better methods for computations) and also in connection with applications. (One can find many, many more similar books. These are simply two of the more recent.)  Now, whether industry will hire a PhD from these areas, I can’t say, but I think we can say that computational algebra and algebraic geometry are active.
A: Just answering to your question in the title: computer algebra is definitely an active area of research! For example, there is a huge research program across several German universities funded by the DFG (German version of the NSF): https://www.computeralgebra.de/sfb/ and https://oscar.computeralgebra.de/. Among all those people involved I’m sure you’ll find someone you know (by name) or who’s doing what you’re interested in.
A: [This is certainly a biased view, but too long for a comment, and hopefully these are some helpful starting points. ]
I'd say it's active but in the US not huge (more elsewhere in the world). A few things to look up:

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*Computational number theory. Of computer algebra/symbolic computation, this may be the area most active in the US (the other areas have some activity in the US, more elsewhere in the world).


*Computational group theory (in the US, a little bit myself, moreso James Wilson, and Alexander Hulpke come to mind, all in sunny Colorado ;), Peter Brooksbank). Dmytro Savchuk @ USF maintains the GAP package for working with automata groups. More generally I'd say look at contributors to GAP, MAGMA, macaulay2, and similar software packages and see where people are (again, most aren't in the US, but some are!)


*Computational commutative algebra / Gröbner bases (Hal Schenck in the US comes to mind).


*The journals suggested in the comments by Peter Taylor are great, but in CS lots of publications happen in conferences (if you're not used to "publication in a conference" or that sounds like an oxymoron...just go with it). e.g. here are some conferences on symbolic computation: ISSAC, FPSAC, CASC, SNC


*There's also probably closely related work happening in automated/interactive theorem provers e.g. the people trying to formalize large bodies of math in Lean etc., but I don't know who/where to point you to for that (hopefully others could chime in).
