Which fields of mathematics are most straightforward for a statistician to do research in, other than probability theory? I have a Ph.D. in Statistics and have always been interested in pure mathematics, but never had a chance to really pursue it. My mathematics background includes real analysis, linear algebra, functional analysis, and measure theory. These were taken at the graduate mathematics Ph.D. level. I am wondering how many classes are necessary for me to take in pure mathematics in order to do research in pure mathematics. Does it require me to go back to school or are there certain areas of pure mathematics that would be easier for someone like me to dive into? Thanks in ahead.
 A: You can start with questions on MathOverflow -- some good places to start are the Unanswered questions in various tags:


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*unanswered questions in classical analysis and odes

*unanswered questions in functional analysis

*unanswered questions in measure theory
Or you can start with questions in the open problems threads; e.g.


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*Is the sequence $(3/2)^n \text{ mod } 1$ dense in the unit interval? Or do all elements of that sequence with large enough index $n$ lie in $(0,1/2)$?


If you have a research background in statistics, and you have a graduate-level background in parts of mathematics, it sounds like you're ready to start mathematical research now.
A: I have PhD in Statistics and work in Machine learning.  I encountered many gaps in the study of positive definite functions/kernels. As a statistician I have very specific questions which can move my field forward if answered but mathematicians never even thought of these questions. What is even more frustrating - I cannot interest any of my colleagues to look into my specific question because they are only interested in working in their own very narrow fields. So I am educating myself in all these  math fields you have just listed.  
