Which are the best universities, worldwide, with the best graduate
level programs in mathematics (pure and applied)? What are their
entrance requirements and how much does it cost?

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Clarification of Question by titromua-ga on 07 Oct 2005 04:59 PDT

If i had to choose, then i would say Chaos and Number theory.

Are you sure it can be broken down that way? There are many
specialities in mathematics, and the experts in each specialty are
going to be different places.

For example, there are approximately 40 experts on Ring Theory
(toplogy subspeciality, more or less)  in the world. Most are in
Russia. One is at Bowling Green University in Ohio.  If you wish to
get a Ph.D. in Ring Theory, you would probably want to go to BG, which
may not have any other outstanding strength in mathematics. I am using
this as an example.

At this level, one needs to find where a Ph.D. is, who can help you
with the specialty you seek, rather than look for "math universities".
There will be no university whose reputation would cover all
mathematical specialities.