Advice about a career interest in Mathematics I am a graduate student and am into 2 years of PhD. My current specialization is in signal processing.
       During this period, in my spare time, i came up with an idea which seems to be a research problem in 
       Mathematics which hasn't been investigated till now in Mathematics literature. I am so much interested in
       this research problem and added to my already existing bent towards Mathematics, i want to study and work
       in Mathematics full time in order to solve this problem, which could give a wide scope to do futher work 
       in Mathematics. I could post this problem as a question on Math.SE, but even if someone is able to answer,
       I think i may not be in a position to appreciate the proof/answer. Moreover I really am interested in solving
       this problem myself. For this purpose I want to approach a Mathematics Professor who could upon looking at the
       research problem would be interested in it (hopefully) and be able to offer me a position (may be PhD or 
       some form of internship) to work under his guidance in order to solve this problem.
In this regard, What are the possible and best ways to approach a Mathematics Professor who would be interested
       in offering me such a role to work in Mathematics. I really do not want to publish/post about the research problem
       in a public forum until I solve it completely. I'd appreciate your advice whether positive/negative in this regard.
EDIT : 
I am currently in Electronics and Communication Engineering Dept. and all my previous studies and degrees are from this Dept.
 A: I don't think you gain anything by being quiet and not communicating with either members of your or other departments or by asking questions outside.  If you are interested in a problem, I would encourage you to attack it in more than one way (that goes for fields of interest as well as in ways of communication).  There are examples of pretty famous mathematicians that started out in EE (e.g., I think Raoul Bott had a non-standard early career  -- an exceptional case, I admit it).
PhD's can be done with advisers in other departments (as long as there is some kind of steady progress and you don't get lost between them).  Transferring departments is sometimes tricky as there is some politics involved (depends on unspoken ranking between departments and how students are supported in each department).  Transferring universities by applying to new programs can put you behind in your career for a couple of years. Not impossible, but at some point you should be capable of just working on problems and you get tired of having to go through different curriculum and prelims.  In our department (physics) at least a third of PhDs have primary advisers outside our department and that includes math, EE, medical, optics, chemistry, geo, CS ....  Getting an adviser or co-adviser in another department is probably the fastest way to PhD (assuming your project goes somewhere and leads to publishable results).
The best you can hope for if you work on your own without an adviser is to publish your work on a field you are not paid to work on.  This is a risk.   At some point in your career (if you get competent) you can work in other fields.  If you are too diffuse in your work you wind up a "dilettante" and this makes it difficult to get funding -- or it's a strength, depending upon opinion.   If nobody notices then it doesn't help you get jobs.  Another reason to get one or more advisers.
I think the overriding thing to keep in mind is you must publish. But it really doesn't matter what in. Some percent of your time can be spent on risky topics.  You can afford to work on hard and long term problems while you are consistently publishing on other stuff.   I spent some fraction of my time during my PhD working on really irrelevant stuff.  But perhaps that time maybe paid off in an ability to attack and make progress in very hard problems later on.  You can budget your time for learning a math toolset or creating a large piece of software in the same way.   If you keep getting results on something else while you are developing or learning then you can afford the time sink.  But if you drop everything and work for 3 years on learning something or developing code and don't get any results on this or anything else then your career is sunk.
