Best schools for each field of study

[toopham]

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

[razor]

You might want to see the US News Ranking for Math Programs. But take them with a grain of salt.

[toopham]

All US rank is top math programs.
they dont go into specific fields. So it seems like ppl think this thread is worthless. oh well.

[enork]

They also have rankings in specific fields for Algebra/Number Theory/Algebraic Geometry, Analysis, Applied Math, Discrete Mathematics and Combinatorics, Geometry, Logic, Topology. To see the full rankings for each you need a subscription, but without one you can still see the top 10.

[toopham]

xD. oops i fail.

[topgun70009]

Bruce Berndt is also a really good (analytic) number theorist, at UIUC. Over 200 publications.

[lambdafunctor]

This is a tough question to answer, mainly because there are too many ways in which to characterize qualitative distinctions between programs, and almost all of them are not accurate portrayals of the quality of graduate study there. Oh, and I'm not going to worry about applied because I'm not sure that I very much care. And I'm basing my commentary on the format of your own post.

Pure

Algebraic Geometry
Stanford - (Ravi Vakil http://math.stanford.edu/~vakil/)... okay, well Stanford is good for AG, but not on the basis of a single individual. If we're going down that road, though...

Columbia- Perhaps the world's most influential group on higher AG; and, of course, Aise Johan De Jong!

Princeton- if you care about arithmetic AG, this is Mecca

Michigan- A number of great algebraic geometers work here, too many to list

Berkeley- same

UIC- It seems that most people who study here study AG

UTexas- AG is a major research area here...

Utah- ...and here.


Miami- Yes, Miami! And Maxim Kontsevich, fields medalist, who works on algebraic stacks.

Algebra/Group Theory

I'll just list the best ones: Harvard, Michigan, Princeton, Wisconsin, UMN, Berkeley, and UCLA.

Number Theory

UCLA -(Terrance Tao)- yes, of course!

UCSD - (Harold Stark)- Mhmm

UIUC - Bruce Berndt- again, fair...

PRINCETON!!!! - And not just for the presence of Andrew Wiles...

Representation Theory

Yale - A lot of the professors there do representation theory. -a silly reason to pick out Yale for RT, though it isn't entirely untrue.

Harvard- some of the most important work in RT has come from this bunch in the past few decades; individuals like Gaitsgory do incredible work here.

Princeton- a lot of work is being done here on the langlands correspondence... fruitful work. It's probably the top RT school right now.


Topology

Harvard- Michael Hopkins, Jake Lurie

CUNY- Milnor!

Stony Brook- Milnor!!

UChicago- Peter May et al.

UPenn- Specifically algebraic topology

Berkeley- Again, the algebraic kind.