Solved – What are important pure mathematics courses for a prospective statistics PhD student

Question

I know that linear algebra and analysis (especially measure theory) are important. Is it helpful to take graduate level courses in real and complex analysis? Should I take courses in abstract algebra beyond the introductory courses, e.g., commutative algebra and algebraic geometry?

Answer 1

In my opinion, some options to investigate at the graduate level could be: functional analysis (a natural framework for statistical formulations), stochastic processes, stochastic control (sequential analysis is optimal stopping), various flavors of PDE (many probabilistic problems are formulated as parabolic and nonlinear PDE's). Pretty much all of these require real analysis at an undergrad level. If you're interested in theoretical stuff, then taking measure theory is also pretty important as a prerequisite for the full treatment of these topics. Complex analysis will have some use, but less than the above; there are connections to probability (i.e. harmonic functions), but it could very well be not worth it

Commutative algebra and algebraic geometry will be not be very useful (one connection I can think of is algebraic statistics, which isn't widely taught). These topics will also be very challenging without a solid background in mathematics.