What areas of pure mathematics research are best for a post-PhD transition to industry? I have a student who is looking to start a PhD in pure mathematics. She is talented and motivated, and will do quite well. She is still in a phase of her development where she is still open to the possibility of working in a broad range of research areas. In order to offset the risk of not finding an academic job post-PhD she would like to write a dissertation that will give her increased likelihood of finding work in industry. She wants to do research in pure mathematics however, i.e. prefers proving theorems to writing code or testing models.

Question: What areas of pure mathematics research are best for a post-PhD transition to industry? Please be as specific as possible.

Answers to the questions here and here are certainly relevant, but these questions are obviously distinct from my question. I think this question is useful to the pure mathematics community in that it addresses the fact that there are so many qualified academic job applicants in recent years. (For this reason, I hope the community gives the question a chance.)
 A: I am not answering your question directly, but will address another consideration which is what is the proper industrial environment to work in.  I have strong opinions:


*

*Never work in a situation where your superior does not have a degree equal to or greater than yours. This occurs often in industry wherein your manager has a BS or possibly MS. The managers are well intended, but they have a significantly different mindset. Not only are they motivated only by money and time but they have very little conception of research and development and the creativity of original thinking. In this situation you will very possibly be considered no more than just another engineer.

*Once into industry, it will be very difficult to enter academia. The longer you stay out of academia, the more difficult. The only exception is if you are one of the especially lucky/gifted scientists who are able to continue publishing and have demonstrated raising grant money.

*If you are not an especially social and outward going person, I would not recommend going into industry. It is just a matter of human nature that people feel more comfortable around someone they socialize and converse with. Promotion in industry has a lot to do with your superiors feeling comfortable with you. Introverts do not get promoted no matter how much superior work they do.


Having said that, it is sad to see how academia has degenerated into a self-protecting tenured focused institution. So maybe industry is a viable alternative. But please take my recomendations into serious consideration.
A: Probability. A strong background in probability will permit to qualify for jobs in
statistics and financial math. See AMS Notices where a lot of statistics about the 
employment of recent PhD is published yearly. And a salary survey.
EDIT. My second guess was combinatorics/coding theory or PDE.
But my friend explained me that PURE combinatorics is not so hot in the (industrial)
job market,
coding theory is not pure math, and pure PDE is very different from numerical PDE,
the last thing is of course in great demand.
EDIT2. Reply on Peter Shor's comment. The distinction between "pure" and "applied" math is not sharp. On my opinion, if a problem arises from a "real world application", it is
applied math (like math physics, control theory, coding theory). If a problem arises
from the inner logic of development of math then it is pure math (like Fermat's last theorem). But of course, the difference is fuzzy, and one can trace almost all math problems to the "real world". Frequently it happens like this: a problem arises in the
real world, mathematicians like it, and start working on it, and then work and work,
forgetting its real world origin. (Example: constructions with compass and ruler, etc.)
Probability was also applied math at its origin.
And it becomes pure math. Similar things we see in coding theory, math physics and computer science.
A lot of "computer science" is pure math.  
A: I'd like to push back a bit on one of the presuppositions behind the question, namely that the best way to "hedge your bets" is to choose a topic whose mathematical content is most likely to be needed in whatever industrial job you might end up taking.  At least in the United States, the dominant factor in getting a job in industry is the impression you make on whoever makes the hiring decision.  For a job where your Ph.D. matters to the employer, chances are that you'll be asked to give a talk about your research, just as you would for an academic job.  Based on my experience, I would say that the key things people look for in your talk are the following:


*

*Can they understand what problem you worked on and why it's interesting?

*Do you seem to really know your stuff and be able to solve problems that others would not be able to solve?

*Are you enough of a generalist that you can communicate effectively with people outside your specialty and pick up new areas quickly?
Note that points 1 and 2 are not that different from what an academic employer is looking for, except that for an audience in industry you will typically need to work harder to achieve them.
With the above in mind, the advice I would give is this: First and foremost, you have to pick an area that you are very interested in.  That gives you the best chance that you will get significant results and be able to communicate them with enthusiasm.  Secondarily, it helps if you can work closely with someone who is not a mathematician, and ideally someone who has experience in industry.  For example, if your advisor has collaborators in other fields then you may want to get one of them on your thesis committee, and get their perspective on what sorts of questions are interesting in their field.  This doesn't even have to be a field that you're hoping to eventually get a job in; the key thing is to accumulate some experience with talking to people outside mathematics and figuring out what kinds of things they find appealing, as well as how to pitch your work in a way that gets their attention.
Of course, professors in so-called applied areas of math are more likely to have the right kinds of connections, so by following my advice you have an elevated probability of winding up in one of the areas that other responders have mentioned.  But my point is that your mindset shouldn't be, "I must learn specific mathematical facts that I will need to know later"; rather, it should be, "I must learn how to communicate with non-mathematicians and demonstrate that I can solve whatever problems they need solved." 
A: You say that your student "... would like to write a dissertation that will give her increased likelihood of finding work in industry. She wants to do research in pure mathematics however ...". The assumption is that she isn't really motivated to work outside academia, and would only take an industry job as a last resort.
That attitude/approach is just not appealing to employers. It's possible to get a job in industry with a mathematics PhD, but you have to want to do it in the first place.  If an interviewer asks why you applied for the job, and you say "Well, I couldn't get a postdoc" then it's game over.
For the rest, I agree with Timothy Chow's answer. 
A: If you know coding you can even escape without even having any expertise (or having minimal knowledge) in probability (like me). I didn't end up into things that involved lots of probability theory simply because I didn't liked it (I got offers from finanical sectors but I rejected the job and preferred to be more into technical area of the industry). If you know how to program then fields such as coding theory, number theory and even algorithmic and algebraic number theory does help. PDE and Analysis obviously helps a lot in the areas of engineering. 
I think the secret is to know how to write a computer program and implement it yourself, because from my experience in the industry: colleagues (esp. from non-mathematical fields) will not believe in your ability until you give them a tangible result and you can only show that something works from a mathematical theory you have in mind if you show them in practice and not if you prove it on paper. This usually involves programming and developing an application that applies your theory, unless you want to end up in a lab and do experiments which is not fun at all. You can almost forget the idea of other colleagues developing an algorithm based on your theories. I remember long time ago I had an idea of a mathematical algorithm that ended up saving my company around 400,000 euros each year but I didn't get any support from my colleagues (who where mostly compute engineers) and so I just programmed my algorithm in my free time and then showed the result when I finally developed the program.
A: Some of my former colleagues (and friends) working in Monte Carlo theory and Quasi-Monte Carlo theory have successfully moved to the banking/insurance sector after finishing their PhD. The first subject could be seen as a branch of probablity theory, the second as a branch of number theory/numerical mathematics. In both fields you can do "pure" mathematics and still successfully convince people that it is very important for applications.
A: I think combinatorics is also good for a post-PhD transition to industry. It is closely related to the computer science. For example, combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of different kinds of algorithms. So when you go to an IT company and you need to write the code or design an algorithm, a good background in combinatorics will help. 
And one part of the combinatorics, the graph theory, also has numerous natural connections to other areas. For example, graph theory gives a model for the network. So I think if you go to the industry to seek a job for companies related the network, having some knowledge of graph theory may be helpful. 
I have to say that maybe combinatorics are not so close to the industry as the probability, since the theorists of combinatorics are mainly interested in the intrinsic property of a discrete structure, and in industry people often use discrete structure such as graphs to solve problems in reality. Though there is a little work to do for a PhD student to find job
in the industry, but a good understanding of combinatorics, I think, will be helpful. There are a few examples of master students and PhD students which transition to industry in our department, so maybe this is a way.
A: If your definition of pure math is "proving theorems", she can work in the following (math related to the following):
1). Control theory (this can mean doing dynamical systems, PDEs, etc): PhDs hired by several leading companies such as Ford, Bosch, Honda, United Technologies, Boeing, NASA
2). Information theory and networks : Internet companies
3). Optimization and/or nonlinear programming: PhDs hired by numerous companies
4). Communication theory: PhDs hired by communication giants
5). Stochastic geometry: Same as 4
6). Probability: Finance/banks/internet
I suggest looking at the top journals of each of these fields, it will give you an idea of the type of mathematics that goes on in there. 
A: As someone who had a PhD in one of the more obscure area in math and is working in industry, I have to say in all of my job finding efforts (software, banks, hedgefunds), the math area that was the focus of my PhD has never come into play. What would have helped me get the job easier ? More programming experience, more statistics, but those aren't pure maths. I also find it hard to believe that specializing in one area of pure math will give you some advantage in solving a real world problem over other areas. Most real world problems don't require knowledge that is terribly deep, it shouldn't take more than a few months to equalize whatever knowledge gap one might have. The rest, which is much more important, is the hands on experience. Would a combinatorics Phd has some sort of inherent advantage in tackling a real world algorithm problem over an algebraic geometry student. I don't think so.
Bottom line, spend few hours a week learning programming & statistics (the applied one, not the theory of statistics). Then still specialize in the area of maths that she really likes.
A: I have thought a bit about this question because I have been contemplating this transition for myself. Here are a few ideas:
-Stochastic analysis: this is a good area of expertise if she hopes to go into finance (though I should include the caveat that recruiters for big finance firms have admitted to me that for recent graduates they filter more by prestige of academic credentials than by research area). There also seem to be many good problems in this area, for instance involving the application of stochastic PDE to differential geometry.
-Machine learning: this is technically an area of computer science, but there are a lot of important theoretical problems and these problems are really math problems (indeed many people working in this area come from a pure math background). You might suggest she take a graduate class in machine learning if her university has one; one doesn't need to penetrate the literature too deeply to find good math problems. This sort of background could help her pursue a career in data science, for instance.
-Mathematical biology: I don't know much this area except that there seem to be lots of jobs for people who know how to model proteins and cells. I've known students who have been jointly advised by a mathematician and a biologist, and this seemed to work out pretty well.
Whatever else happens, if she is serious about this I would strongly encourage her to learn how to code if she hasn't already. I think there was a time when a mathematician could reasonably hope to get a job designing models or algorithms for engineers to implement, but that time seems to have passed. At the very least it will make her a much more flexible and attractive applicant.
A: 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.
A: A PhD at the interface between logic and theoretical informatics (for example Lambda calculus) is pure mathematics, and surely interests fashionable companies like Google or others... 
