Any risks down the line choosing maths vs. stats PhD programme?
Provided one can find a supervisor who has the background and research interestes that would allow one to keep extending ones pure mathematical toolbox are there any major risks in going for a PhD in stats?


There's a small risk, but it can be managed.  The first issue is that you need to make the mathematical content of your work very clear, for example by publishing in journals that could be considered both math journals and statistics journals (e.g., IMS journals).  However, if you're interested in math departments I assume you'd be doing that anyway.

The slightly more subtle issue is how mathematicians view statisticians.  There's sometimes a mild prejudice that people in mathematical statistics are overly specialized and outside of the mainstream of mathematics.  For example, it's possible to get a Ph.D. in mathematical statistics while having considerably less breadth of mathematical background than would be considered acceptable for a math grad student.  (Of course, the flip side is that you are expected to know other things instead.)  There's sometimes a fear that a statistician would be unlikely to talk much with other math department members, or might be unwilling or unable to teach anything but statistics.

Plenty of statisticians have found jobs in math departments, so I don't want to be discouraging.  However, I'd recommend focusing on mathematical breadth.  For example, if you work with people in combinatorics or algebra, then it will be clear to everyone why a math department is a natural fit.  If you talk only to statisticians, it will be less clear.  It can still work out even then, but generally when the department either has a thriving statistics group or has decided they really need a statistician (and either way this cuts down on the flexibility of your job search).