This is a query aimed at those who have their doctorates in math.
May I ask what your theses were on?. When one goes for ones doctorate, what does it entail as compared to attending classes for a bachelor's and/or graduate degree?. Do you mainly just choose an area of interest and research that for your doctoral theses?. I am just curious.
I do not have doctorate in Mathematics - I did it in Engineering (Mechanics). In there, I first chose the "project" that was willing to support me (through assistantship) and then I drilled in further to choose the topic of my dissertation (this is the 90% of doctorate degree - at least in my case). I had to complete 60 hours of post-graduate course work - but that was piece of cake.
Depending on your advisor and doctorate committee (I had to choose those) - you will be required to write 2 - 4 papers in "reviewed" journals. That could be the biggest (hardest) requirement for Ph.D. However, this requirement is generally "unwritten". You can generally write your thesis weaving these papers into one tome (or you could be like Einstein - and write 3 papers in 3 different topics and each one noble-worthy and none were for his thesis).