Are there any research careers except professorship for a person holding PhD in pure mathematics?

I'm interested in mathematical research, but want to know if there are good jobs besides being a professor. Are there any lucrative research-centric careers out there for someone with a PhD in pure mathematics?

Answer

I can speak from personal experience. I have a math Ph.D. on a topic with absolutely no real world applications, I work in industry, and my job title is actually "Senior Research Expert". So yes, there are research careers in industry. And while I won't get rich off it, I can't complain about the remuneration.

However, (almost) nobody in industry will pay you for thinking about stuff, going to conferences and writing papers. My research is extremely applied, and I am very much constrained by a) what customers will pay for and b) what is feasible given our code base and software architecture. I simply can't go off on a tangent and argue that a particular algorithm is very elegant, if it is not implementable in a reasonable amount of time or nobody will buy it.

Depending on your Ph.D. topic, be prepared to change fields. I used to do something like discrete optimization - now I do statistics and time series analysis.

While I personally don't code (which the developers are very happy about), I work closely with the developers that actually turn what I thought up into software. Interpersonal skills and the ability to fit into a team and a software development process (Scrum) are much more important here than in academia.

Research mathematicians are rare, and few people understand what I do and why I'm paid. It took a long time and some very fortuitous circumstances for me to demonstrate that I do add value to the company, and I was lucky to keep my research job when my company was acquired - my new employer really doesn't have research positions as such, so I had to justify why I should keep my niche.

Finally, there are about five other originally pure mathematicians in the team I work with, two of them with Ph.D.s. None of them do anything that could be called "research". They develop software or do analytics.

Summarizing: yes, there are math research careers in industry. They are few and far between. Be prepared for an uphill battle, to change fields, to do what needs to be done.