I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.


I'm interested in both algebra and algorithms. So I read a little in some computational algebra books (comp. group theory to be precise) and it looks like a great choice for me.


My questions:
1. Is this a "hot topic" in mathematics? Are there many serious people researching it?
2. Where are the major places in the world to research comp. algebra and to get a PhD?


Thank you!



Computational algebra: where?
The first piece of advice I would give you is not to prematurely limit yourself to a narrow area, unless your M.Sc only takes a year. If it takes at least two years, then you will still learn plenty of exciting mathematics before you have to make a reasonably definitive choice. If you are thinking of doing a PhD in the states, then even the PhD itself is so long, that being at a generally strong department might be more beneficial than to have an isolated area of expertise around you.


Having said that, if you are interested in computation mathematics of an algebraic nature (computational group theory, number theory, geometry, more general algebra), then a natural choice would be some place where one of the big computer algebra packages is being developed. The sites I list below also provide a good list of contributors, which will help you in choosing a shortlist of possible advisors. Some examples include


1. [MAGMA](http://magma.maths.usyd.edu.au/magma/htmlhelp/ackn.htm) in Sydney
2. [GAP](https://www.gap-system.org/) in various places; this one is particularly focused on group theory
3. [PARI/GP](http://pari.math.u-bordeaux.fr/) in Bordeaux
4. [SAGE](https://www.sagemath.org/) in Washington, whose site also provides [a nice map](https://www.sagemath.org/development-map.html) of where the developers are based. Thanks to Suvrit for fixing this unforgivable omission.


If I can think of more, I will add them here, but these should give you a good place to start. You can also browse through some of the dedicated journals, like the [Journal of symbolic computation](https://www.journals.elsevier.com/journal-of-symbolic-computation) or the [LMS Journal of computation](https://www.lms.ac.uk/publications/jcm) and see who the prolific contributors are. Reading some of those articles might also help you decide whether this is what you want to do.


As for your question whether this is a "hot topic", that's harder to answer. It certainly is in the places mentioned above and in various others, where there are strong computational algebra groups, but I know of some places where this area of mathematics is frowned upon by the more theoretical community. But I suppose that that's probably true of almost any area of mathematics and you shouldn't let that deter you.


