Math Graduate Schools
Hi. I'm a junior math and physics major. I'm hoping to go to grad school for a Ph.D. in math, and I'm trying to figure out some good schools to apply to. I found the math subject I enjoyed the most was functional analysis. I did a basic course in it using a book called Introductory Functional Analysis by Kreyszig and it was one of the most enjoyable math books I've ever read and the problems were pretty fun. However, I'm not too sure exactly what schools I should be looking at for grad school. Does anyone have any suggestions of schools that have a good group of people specializing in functional analysis or operator theory? Also, if anyone has any recommendations for books a bit more in depth in the subject, that would be cool.
Thanks! 
Re: Math Graduate Schools
Try looking up some of the more renowned analysis departments, this is all subjective of course. But off the top of my head, I think UCLA has a well regarded analysis department. I know Texas has a very good analysis department (for PDEs anyway).
I don't have the link with me, but try looking for the US News grad school rankings. They rank math grad schools by specialty, so check out the analysis rankings. Of course, this should only serve as a rough guide, not a definitive source. Some schools have excellent advisers but don't have the overall strength in their departments. With those analysis rankings, check out the departments, see how many specialize in functional analysis, how often they publish, do they advise students on a regular basis, etc. Also, look for a functional analysis math journal (there's gotta be one right?) and lookat who publishes and what they are publishing. This is a good way to spot potential advisers. Once you have found some prospective departments, then you need to consider location, postdoc placement rate, percentage of students who complete their thesis, like the typical questions you ask about any grad school. But to me, the most important thing should be that they have some potential advisers who are working in subjects you like. Then you can worry about nonmath stuff. Finally, just to add, have you read Barry Simon's 4 volume text on functional analysis and mathematical physics? Simon is a top notch functional analyst over at Caltech. Check him out, he does functional analysis, Schrodinger Operators, etc. Best of luck! 
Re: Math Graduate Schools
Analysis rankings:
1. Princeton 2. Berkeley 3. MIT 4. UCLA 5. NYU 6. Chicago 7. Harvard 8. Wisconsin 9. Michigan 10. Texas Don't take these too seriously. Caltech isn't listed, which makes this listing rather dubious, seeing as how their entire department seems to revolve around analysis, and they're somehow ranked 7th in the general ranking. 
Re: Math Graduate Schools
Geting into the top 10 schools will give you much more connetions and slighly more credibility.

Re: Math Graduate Schools
Thanks for the replies thus far everyone, and thanks to JasonJo for the book recommendation, I'm gonna try to pick that one up in my library. Although top 10 programs are nice, I'm more interested in top 50s, as I don't want to gamble and just apply to top 10s! Also, I checked out the people and sites for some of those top 10 schools, and they may have had a good number of analysts, but not too many of them seemed to be in operator theory or functional analysis. Texas (Austin) looked pretty cool though, and I've heard good things about that place.The only school I've found so far with a substantial number of operator theorists is Kent State, but I'm not too sure what that place is like. I also saw that U Maryland, College Park seems to have a good number of faculty in the area I'm interested in, and I think that's a well regarded school, so if anyone has been there and has opinions on it, I'd love to know!
Thanks! 
Re: Math Graduate Schools
Talk to some of the analysts at your school .. perhaps the professor you took the course with can give you some insight.

Re: Math Graduate Schools
EbolaPox
Here are a few more good functional analysis and operator theory groups that haven't been mentioned yet: Vanderbilt http://www.math.vanderbilt.edu/~ncgoa/people.html Texas A&M http://www.math.tamu.edu/~kdykema/LAN/lanpeople.html Waterloo (Canada) http://www.math.uwaterloo.ca/PM_Dept...chareas.shtml Toronto (Canada) couldn't find a website Leeds (UK) http://www.maths.leeds.ac.uk/pure/analysis/ Also see: http://www.amsta.leeds.ac.uk/nbfas/ There are plenty more. Here are a couple of websites that may be of interest: Operator algebra resources http://darkwing.uoregon.edu/~ncp/OpA...ges/opalg.html Banach algebra resources http://www.math.ualberta.ca/~runde/ba.html As for books, for something to read after Kreyszig, try Conway or Pedersen, although unfortunately they are not as nicely written. For operator theory and operator algebras, there's Kadison & Ringrose's two volumes, Davidson's C*algebras, Douglas's Banach Algebra Techniques, Arveson's brisk Invitation and his equally brisk Short Course on Spectral Theory, etc. A good, free book is Sunder's Functional Analysis, available at his website: http://www.imsc.res.in/~sunder/. 
Re: Math Graduate Schools
I'm not a functional analyst but if you had some exposure you cannot go wrong with Rudin? I also heard some good things about Lax, but it doesn't seem to be that popular.

Re: Math Graduate Schools
Good info everyone, thank you! I didn't know about Vanderbilt, so I checked it out and it looks good. I'm also glad Texas A&M is good, as I got an REU there. Thanks for the operator algebras resources and links!
I will be checking out Conway's book and Rudin's book, as recommended. Thanks again everyone. 
Re: Math Graduate Schools
I have recently got admissions into Texas A&M and University of Florida. I am awaiting responses from Washu in St. Louis, Utah, SUNY Buffalo and Waterloo in Canada. Do you have any idea where the above places stand relative to each other? In particular, between Texas and Florida?

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