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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!

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    $\begingroup$ If you want to transition to industry, why do the PhD? $\endgroup$
    – user44143
    Commented Aug 27, 2021 at 15:52
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    $\begingroup$ @MattF. I find the process of doing a PhD rewarding. $\endgroup$
    – Johndoe
    Commented Aug 27, 2021 at 15:54
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    $\begingroup$ @Johndoe: If you don't get useful answers here, try asking on Computer Science, under the tags "computer-algebra" and "mathematical-software". But do wait for a few days before asking the same question on another site ("due dilligence") - it's one of the unwritten rules of the Stack Exchange sites. $\endgroup$
    – Alex M.
    Commented Aug 27, 2021 at 16:14
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    $\begingroup$ I don't know how mainstream it is, but certainly RISC Linz seems to still be active, and there are a good few journals publishing in the area. $\endgroup$
    – Peter Taylor
    Commented Aug 27, 2021 at 16:43
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    $\begingroup$ Just to add five cents to the discussion, many techniques from symbolic computation and computer algebra are useful in verification and/or computer-assisted theorem proving, so you might want to look at those areas too. $\endgroup$
    – xuq01
    Commented Aug 27, 2021 at 19:52

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[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:

  • 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).

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    $\begingroup$ Just the standard disclaimer: Lean isn't an ATP, but an interactive theorem prover, with some small amount of automation. Syncing up formal proof assistants with symbolic or computer algebra packages is an area ripe for picking, since it would help to have formally verified computations, and also formal proofs that have access to efficient algebra algorithms. $\endgroup$
    – David Roberts ♦
    Commented Aug 28, 2021 at 11:10
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    $\begingroup$ @DavidRoberts: I don't know about "ripe for picking". The fruit's been hanging on the tree for more than 20 years. $\endgroup$
    – Andrej Bauer
    Commented Aug 29, 2021 at 8:31
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    $\begingroup$ Of course, I was rather trying to say (in a bad way) that even though people have been interested in the combination of symbolic computation and formalized mathematics, for some reason the imagined benefits have not really been achieved. $\endgroup$
    – Andrej Bauer
    Commented Aug 29, 2021 at 13:44
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    $\begingroup$ I think it's mostly social reasons. A type theorist feels nausea when they see how badly unsound CAS are, but does not know how to fix them. An expert on symbolic computation feels like type theory is just an enormous complication that hinders real work. I have worked on both sides, each knows very little about the other. $\endgroup$
    – Andrej Bauer
    Commented Aug 29, 2021 at 21:08
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    $\begingroup$ +1 to what @AndrejBauer writes -- and as someone working on CAS, I am actually keenly interested in formal verification, sound type systems etc. -- but as it is, I am fighting an uphill battle just to get everybody to at least put references in comments next to the algorithms they implement. Of course often enough no references even exist; or the implementations differ substantially from the papers due to "obvious" optimizations (which often are buggy; but equally often essential to make the algorithms effective), etc.. I hope we'll get better at this eventually. $\endgroup$
    – Max Horn
    Commented Aug 31, 2021 at 20:53
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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.

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Two books and a journal:

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

  2. Applications of Polynomial Systems by Cox et al.

  3. 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.

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