3

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.

astatistician
  • 31
  • 1
  • I am not qualified enough to fully answer such a broad question. You could perhaps employ your expertise in statistics to help bridge the gap with applied category theory: an example is this preprint by Rémy Tuyéras. – Aurelio Jun 08 '20 at 15:43
  • 1
    I suggest looking for research on rigorous mathematical foundations of statistics that you find interesting and are not too far of a leap from what you already know. From there you can probably gradually migrate into research in areas such as probability and the others mentioned in the answers. But it's best to start with something that has connections to your current area of expertise. – Deane Yang Jun 08 '20 at 17:43
  • Perhaps ergodic theory as a bridge from statistics to dynamical systems (or many other fields, ergodic theory has numerous applications and links). – John Coleman Jun 08 '20 at 23:47

2 Answers2

3

You can start with questions on MathOverflow -- some good places to start are the Unanswered questions in various tags:

Or you can start with questions in the open problems threads; e.g.

  • 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.

  • Do you really believe that such a famous hard question (say the $(3/2)^n$ problem) is a good starting point? – YCor Jun 08 '20 at 17:53
  • 2
    That depends on the OP's goals. If the goal is a stream of publishable papers, no. If the goal is fun, or a goal that will motivate learning more math, or just the experience of research in a new area, yes. –  Jun 08 '20 at 17:56
2

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.

Tanya Vladi
  • 168
  • You may be able to rephrase your questions so mathematicians find them more intuitive -- I rewrote the title of one of your questions here in hopes of doing that: https://mathoverflow.net/questions/361587/smallest-s-such-that-x-i-x-j-delta-xi-i-xi-j2-0-delta1 –  Jun 08 '20 at 17:25
  • @MattF. thank you. – Tanya Vladi Jun 08 '20 at 19:09