I once did most of a PhD, and bailed out to take an industry job, and I'm still commercial, although maths is my hobby. That was around twenty years ago.

I'm a contractor/consultant, so I've been to *loads* of interviews and interview-like situations. As far as I remember, no-one has ever asked what my PhD was in, or even cared about the fact that I never finished it.

What they do care about is the sorts of programs I can write, and occasionally how good I am at solving toy problems. Mostly they look for ability in languages like C or python. When the people who are offering the jobs are themselves computer science graduates or very good programmers, they care about whether I can write in one of the functional languages, say Ocaml, Haskell, or Lisp (any one will do, or anything similar).

Mathematicians are always good at solving the sorts of toy problems you get in interviews. I'd recommend 'Are you smart enough to work at Google' for lots of examples of the sorts of things interviewers ask. She'll read it in an afternoon and enjoy it.

Writing programs tends to be seriously illuminating when you're trying to understand a mathematical idea.

So I'd advise her to do some piece of maths which deals with problems where you can use a computer to get answers. (I'm pretty sure that's all areas of maths, actually, but maybe there are some subjects where a computer would be useless. In which case I'd be worried that they weren't actually about anything.).

And she should try to write programs and make pictures about her field. And she should write these programs in C, python, and lisp, to get the feel for the differences between them. Mathematicians tend to love lisp. My favourite flavours are scheme and clojure. If she has any matrices to multiply, she should try MATLAB/Octave as well.

She should also play with maxima/mathematica/maple if she does a lot of symbolic calculation, and R if she does statistics or likes to make nice graphs.

I think if she does this she'll have a serious advantage over non-programming mathematicians in maths itself, and if she eventually goes into industry, she'll have picked up the crucial skills by magic, and have enjoyed the process.

As far as which area to choose goes, a Maths PhD is such a dreadful experience if you don't love your subject that the only possible advice is 'either do something you're obsessively interested in, or don't do one at all'.

If she has more than one thing she's obsessed about, pick the one that's more amenable to understanding by writing programs.