I dont see this thread anywhere and I believe this thread should be a sticky like the admission result thread.

So I plan to list all the fields and the schools that are best known for the field. I am just 1 person so my knowledge is very limited.

And sometimes I can be wrong so if you think my data is incorrect then please tell me and I will update the list.

If you know that of a school that is very good in a certain field of study then please post and I will update the list.

Applied

Mathematical Physics

Caltech - Physics is big there.

UC Santa Barbara - Physics is big here too.

Fluid Dynamics

Optimization/Operation Research

Probability Theory

Stats

University of North Carolina, Chapel Hill.(its just in my head that this school is big for stats. I dont have a good reason why.)

Financial Mathematics

Game Theory

Discrete

Combinatorics(Graph Theory, permutation, enumeration)

MIT- (because of Richard Stanley)

UC Davis- (Fu Liu, student of Richard Stanley. I might be biased here but I think she's amazing.)

Pure

Algebraic Geometry

Stanford - (Ravi Vakil http://math.stanford.edu/~vakil/)

Algebra/Group Theory

Number Theory

UCLA -(Terrance Tao)

UCSD - (Harold Stark)

UIUC - Bruce Berndt

Representation Theory

Yale - A lot of the professors there do representation theory.

Topology