For graduate school the decisions come from the respective departments, especially for the technical subject.
One ray of light though since you said you have money you may be able to get in but required to pay your own way, an option that is not suitable for 99% of the rest of the applicants.
Just playing around, no easy choice and it depends 100% on your values, if you value the princeton degree highly (as in 15,000 per year more) then go for it.
Personally I think Berkeley is the way to go but that's because 15k a year sounds pricey.
Also I am sure if you kick *** in your...
For jobs they will be very interested in what software/programming skills you have (SAS ,R etc), projects you have worked on. Which ever courses let you become more comfortable with computing will be better job wise. Especially if you can link the skill with some database stuff.
Why not do a PhD in finance? Do you want to prove things like central limit theorems (and generalizations of that) or do you want to understand them and use them in finance.
If you want to use them then do PhD in finance if you want to prove them then ...
to improve your ability to do...
Doctor, you can always come back to physics. If you want something in between do physics with a view to eventually do medical physics. But I say go for medicine. Sometimes the subjects are not as "fun" anymore, the environment for work is sometimes poor (takes long to get equipment you need or...
Math is too huge a field to answer, maybe if you narrow your interests you can get more suitable answers.
You have people who did undergraduate studies in econ, comp sci, physics who pursued a phd in math. As Lateraius said you need to look at programs so you get a feel for their expectations...
For industry I think physics is probably just as useful as number theory (out of the frying pan into...). Number theory in it self is a large field but for the type of person who was interested in that field I would suggest learning stuff on optimization in particular "integer programming" that...
https://www.amazon.com/dp/0387989803/?tag=pfamazon01-20
They take for granted some things about manifolds, (thats from differential geometry)
They review a lot of the basic commutative algebra making it "ideal" for someone cutting their teeth.
There is another good book mentioned above...
There was a newspaper article about him which you can read for more info.
http://weblog.science.fau.edu/info/images/israel.html , since this is a write up that says how typical this situation is. More on FAU someone there did pretty well in the Putnam exams...
Technically there are not many formal prerequisites, in fact I think a lot of books claim to start from scratch. The bible (Hartshorne) only formally requires Atiyah Mcdonald.
However to get the right intuition for Alg Geom in hindsight I think the best way to approach it is to try and know...
It also how do we define easy, as we sometimes mix tedious with hard. Sometimes these intro courses are just a big con to get you to like math, then when all the fun stops it is usually too late. What books where used for these courses( if I may ask)?