I am applying for graduate school in pure mathematics and I recently got very interested in C*-algebra.

I am definitely wrong but I get the feeling that C*-algebras is not as popular as other areas of pure mathematics like number theory, analysis, algebraic geometry, etc. It also seems that most top ranked universities like MIT, Harvard, Stanford, Princeton, etc do not have any active research group in C*-algebras.

If my observations is right, then what is the reason? Is it because C*-algebra is harder than other areas of pure mathematics or is it because it is still a young area of pure mathematics?

Given that I am interested in C*-algebras and most top ranked universities are not active in this area, where can I apply? Also, will doing graduate work in C*-algebras instead other more popular areas of pure mathematics have a negative effect on my academic career?

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    $\begingroup$ Both UCLA and UC Berkeley have strong groups working on operator algebras and related topics. And there is this practical site that allows you to search for operator algebraists: operatoralgebras.org/directory.html In general, I think your comparison with other fields is off - number theory or algebraic geometry are just much broader than $C^\ast$-algebras. $\endgroup$ – MaoWao Jan 3 '20 at 12:24
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    $\begingroup$ I would be interested to know if there is any truth in the OP's impression. As a quantum physicist I always thought C* algebras are an important part of modern math and even more so in the '30s- up until probably '50s-'60s. Am I wrong? $\endgroup$ – lcv Jan 3 '20 at 12:41
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    $\begingroup$ I think the OP's impression is indeed wrong. There are several groups worldwide working on operator algebras with a focus on C*-algebras: in Germany for example there is a big group in Münster and there are groups in Göttingen and in Erlangen (the latter with a focus on representation theory and on topological insulators). In the UK there is Oxford, which recently hired a professor working in the classification programme of nuclear simple C*-algebras, Glasgow has a big group in operator algebras. The Newton Institute at Cambridge had a whole programme on operator algebras in 2017. $\endgroup$ – Ulrich Pennig Jan 3 '20 at 13:21
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    $\begingroup$ @UlrichPennig I think (but have not really tested this theory against the evidence) that because of the tenure system in North America the Elliott-Toms-Winter-fueled resurgence has been slower to translate into new hires than in Germany, Scotland or Wales (also waves to Xin at QMUL). She is correct to note that at MIT, Stanford, Harvard op alg is not a thing; while at Berkeley aren't they down to Voiculescu and Jones as emeritus? $\endgroup$ – Yemon Choi Jan 3 '20 at 15:56
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    $\begingroup$ @NikWeaver Well I'd forgotten about Marc Rieffel, tbf, but yes time has done its thing... $\endgroup$ – Yemon Choi Jan 3 '20 at 16:07

One way to tell how active a field is is by looking at what's appearing on the arXiv in that area. I think that will show you that operator algebra is a robust subject with a lot of activity.

In the comments, MaoWao points out that UC Berkeley and UCLA have very strong operator algebra groups, and Ulrich Pennig mentions groups in Münster, Göttingen, Erlangen, and Glasgow as places with substantial groups. Copenhagen is another good example.

On the other hand, the OP's observation that most top schools don't have an operator algebra group is quite correct. I would think that simply has to do with the size of the field --- there aren't enough C*-algebraists to populate that many departments. The two Fields Medals in the subject (to Connes and Jones) show that people in other areas do respect the field, I think.

The question is partly about career advice. All I can do there is report my impression that C*-algebraists don't seem to have more trouble finding employment than mathematicians of equal ability in other areas.

You ask which schools you should apply to --- in the comments MaoWao gave this site, which I was not aware of, which lists operator algebraists worldwide. It looks pretty complete to me.

However, my last comment is that as a prospective graduate student, you are at a very early stage to be settling on a specialty. Not saying you shouldn't, but I think most of us would recommend putting your main effort into getting a broad education during the first two years of grad school.

Related MathOverflow questions:

What are the applications of operator algebras to other areas?
States in C*-algebras and their origin in physics?
Quantum functional analysis
applications of C*-algebras in the field of PDEs

  • $\begingroup$ Oh, one other thing. You ask which schools you should apply to --- if you're in the US, Berkeley and UCLA for sure, those are the top ranked schools, but there are plenty of good places with more than one operator algebraist. I don't know of any sort of directory that would tell you where, but just looking at any particular department's website should tell you in most cases whether they fit the bill. $\endgroup$ – Nik Weaver Jan 3 '20 at 14:55
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    $\begingroup$ Regarding a directory of operator algebraists, there is the one I linked in a comment above: operatoralgebras.org/directory.html But I only stumbled across it by chance, so I don't know how complete it is. Do you happen to know (at least you are included)? $\endgroup$ – MaoWao Jan 3 '20 at 15:11
  • $\begingroup$ @MaoWao: oh, I didn't see that! I'll add this to my answer. $\endgroup$ – Nik Weaver Jan 3 '20 at 15:15

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