Research topics restricted to students at top universities?   Hello everybody.
I am a Ph.D student in North America looking for advice about my prospective research area.
My supervisor works in a research area, let's say area A, so as soon as I was accepted as his student I started to learn the background in the area A. At some point I started to study by my own the connection of area A with a nearby area, let's say area B. I found that area B match much better my interest, I can use my former background in area B and definitely I have more mathematical intuition in that area.    
I have spent the last six months studying really hard several advances books, recent papers and lectures notes to get the necessary advanced background to start doing research in area B. Some days ago I had a meeting with my supervisor, I was really enthusiastic explaining the connection of area B with what he does in his research (area A), however my supervisor told me that it is not a good idea to continue working in that direction because it is difficult to be accepted in the circle of people who work in area B (several professors at top U.S. universities and their former/current students), that there is a lot competition between them to publish results and because neither my supervisor nor me belong to that circle it would be difficult to publish or even find a specific problem for my research.
I did not understand what my supervisor means when he said " it is difficult to be accepted in the circle of people who work in area B", does that mean that if I submit a paper it would not be published even if it is correct, well-written and meets the standard of quality and originality of the journal? or if I try to submit a talk to a conference in that area my talk would be always rejected?
I find difficult to believe that there are areas of mathematics that are closed to people who does not belong to a certain circle of leading researchers and their students.  
Despite this, I consider that is possible to perform my research in area B, I could contact by email people working in that area, several of them posed open questions and further directions of research in recent papers, moreover at the end of this year I plan to attend an important conference specialized in area B, a lot of junior and senior researchers of "the circle" are attending and I am interested and familiarized with the research of several of them, so the connections I could make there would be very useful.  
To sum up my questions are:
i) Is is true that there are areas of mathematics that are closed to people who does not belong to a certain group of professors and students at certain universities?
ii) If I make the right connections with people working in area B, do you think is feasible to perform my Ph.D research in area B with my supervisor, whose research is just nearby area B, and the advice of a specialist in area B mainly by email?
Some additional information: I am not very far from those universities where the people of "the circle" are, so I could travel from time to time to those universities to meet those people or give talks at the seminars. 
Thanks for your answers.
 A: I have seen cases where the student and supervisor were in different places. They worked badly.
A: If area B can be applied to area A as you imply then I would suggest trying to do that. Your advisor should be able to tell if what you are doing is genuinely new and worthwhile in this case. 
A: Perhaps your advisor had meant merely that the group who work in area B are very strong and have a rich knowledge, and it is difficult for outsiders to enter into or compete with that group because they won't have risen to the high expertise to which that group had brought itself? You seem to present the issue as one of political intrigue and exclusion, but it may not be like this at all. There are surely many mathematical groups, who by working intensely on a focused topic bring themselves to a high level of expertise on that topic. If this is the situation, then it would seem by your other remarks that you can make contacts with that group and begin to study with them and thereby involve yourself in their expertise. 
That said, I also believe that it is wise to listen to one's advisor's suggestions about topics of investigation. It may be that your advisor simply feels that he will not be able to help you as much in area B, simply because he doesn't himself have the knowledge necessary to guide you in that area. Thus, your plan to work in area B is essentially amounting to not working with your current advisor, and instead having only an email advisor, who may not ultimately give you the attention that you will want and need later on, and that may not be the best situation. But if there is someone in that group who can server as your mentor, then it may work out.
Regarding questions (i), I think this kind of concern is likely misplaced. In my experience, any mathematician with talent will eventually be recognized for it, regardless of whatever connections they may or may not have.
A: Another consideration that your adviser might have had in mind is that the experts in area B might be slow to publish or even to write up their work, so that the literature doesn't reflect the state of the art.  In that situation, you might find yourself rediscovering what they already know but haven't published.  I think this difficulty can be minimized, or even removed altogether, if you can establish really good communication with the people in area B, so that they tell you their latest discoveries, even if they haven't yet written them up.  But establishing such a level of communication may take considerable effort on your part.
A: I hate to break it to you, but academic publishing can be political. The political component is smaller in math than in other areas, but not zero. 
Here is a study that shows articles previously published by esteemed psychology researchers were rejected when re-submitted under a different author's name, and not simply because reviewers recognized the articles. The papers were subjected to greater criticism when given a different author.
I don't want to exaggerate the effect of prejudice, just to say it cannot be ruled out.
A: I'd very much encourage you to contact one or more of the leading researchers in Area B and ask for their advice.
A: I know of at least one field that matches your description of Area B, so your advisor's advise may not be ridiculous.
A: Having very limited knowledge of the specifics it is impossible to know what you should do in your particular case, or whether your advisor is correct in sensing danger.  Some fields have harder competition, harder problems, and less low-hanging fruit to collect.
However, there are a few things which might not occur to a beginning graduate student:
A.  There is nothing that prevents a student at one university from working formally or informally with professors at other universities.  If your current advisor interferes with this you have the option of finding another advisor.  You might even be able to relocate to the other university, whether or not you keep your official status as a student at your current university.
B. There is nothing that prevents a graduate student from submitting independent results to a journal, giving talks on said result at conferences, or being hired based on such a result.  Reading the current literature is a good way to find problems that interest you that also interest other people.  Solving someone's published conjecture or extending their published work as a graduate student is a good way to get hired by that person as a postdoc.
C. The goal of grad school is to leave as an independent researcher with your own research momentum and the ability to find your own problems and research program.  It is a very good sign that you are developing your own interests. 
D. It is much harder to work on what someone else is interested in than what you are interested in.
E. If you go to a conference in area B then you'll meet all the professors there.  They won't care if your current advisor thinks you should work in A rather than in B.  By all means you should go to any conference that interests you.
F. Professors do not always agree with each other, even if one happens to be your advisor.
Whether any of this is relevant to your situation is unclear.  Is area B simply an over-competitive shark-tank?  (If so, do you feel like shark or like "bait"?)   
A: (i) Is is true that there are areas of mathematics that are closed to people who does not belong to a certain group of professors and students at certain universities?
Theoretically nothing is closed to anybody. Practically, if  that group is responsible for some of the most important development, then  a beginner in the field, with  limited interactions   with the leaders has only very small odds  of discovering something  new.
(ii)  If I make the right connections with people working in area B, do you think is feasible to perform my Ph.D research in area B with my supervisor, whose research is just nearby area B, and the advice of a specialist in area B mainly by email?
Theoretically yes, practically difficult.  The amount of insight you can get by e-mail is limited. Also, the  time  that a non-adviser can spend with you is rather limited.
Because I don't know you I am hesitant to advise you. I can only  speak  of my experience. I have always lived between several worlds. When I started I was  about 60% in A and 40% in B. It worked out  fine, but   I had some challenges.   However I always enjoyed the experience.
A: I (LSpice) have copied and pasted Matt Emerton's excellent answer from the comments (1 2 3 4), which seems OK (with attribution) since this is community wiki anyway.

There are some areas of mathematics where the research is at a very high level, and where there are a fairly small number of leading experts who know each other more-or-less personally, so that ideas are shared not only through published literature but through more informal channels as well. In these circumstances, it can be hard to "break into" the area from the outside. 


The most basic reason is that in such situations one can find one's hard work being trumped by more general and powerful results that suddenly appear from the core group of researchers. Students who are working with the core group are hopefully spared this, b/c their advisor knows what everyone in the group is doing (more-or-less), and can try to protect their students a bit from being crushed by the other leaders in the field (through careful choice of problem, through communication with the leaders and letting them know what their students are working on, etc.). If your advisor is not connected to this core group (and it sounds like they are not) then they can't offer you this mantle of protection, and this sounds (to me) to be in part what your advisor is alluding too. 


I would ask your advisor to elaborate on/explain their comments (since it seems that you haven't fully understood whatever point they were trying to make). Also, if one of their concerns is the one I just described, this can be alleviated to some extent if you build your own personal connections with the area B core group. 


Just to conclude: math doesn't have to be a competition, but in old, highly developed areas, with a strong group of leaders who have a good grasp of where the field is moving and what results are and aren't currently in reach, it can be difficult for a newcomer to navigate and avoid having their work be literally trumped or at least overshadowed by the work of the core group. Since your advisor has explicitly said that area B is competitive, it seems that this is one of the concerns they have. So if you do move into area B, you should at least bear this in mind. 


One way to deal with this situation is to begin working on areas close to, but not exactly in, and not quite as mathematically central, as the key concerns of area B. This can provide you a chance to develop your technical skills, and then you can try to move into area B from a position of mathematical strength, with some solid experience already under your belt. I know several people who have done this successfully.

A: I will try to point something complimentary to the existing answers.
The problem here is not that "your advisor is not interested in you contributing to area B", and also the problem is not that "only elite groups works on B". The problem is your university and the phd program which brings constraints on your long term schedule and graduation. So work in "A" to finish phd, remembering "every area offers challenges in its own right and A is no different". After graduating, get a tenure or whatever that is, to have freedom to work on area of your choice. Keep this strong in your memory, till you graduate.
