I'm interested in optimization generally. I don't know enough to break it down to sub-fields that I would work on.
Probability and statistics seem relevant. What about calculus of variations/functional analysis? Can anyone explain how they fit into the bigger picture (if at all)? What else should I know about? My school does not have a class on functional analysis, but on "Banach Algebras and Spectral Theory".
I recommend a course in linear programming after taking linear algebra. Take a look at "Introduction to Operations Research" by Hillier and Lieberman. A similar book is written by Wayne Winston. I also recommend reading the introduction to "Optimization by Vector Space Methods" by Luenberger if you are interested in how functional analysis relates to optimization.