How can I get a PhD position to work on solving a particular mathematics problem that I have formulated independently?

Question

I have a pure mathematics problem which I formulated after years of research in signal processing of audio signals and years going into formulating a problem on precise mathematical terms, which is a provable/disprovable mathematical statement. (I avoid using the term conjecture as I am not qualified enough)

My belief is that it has very important applications in real world signal processing. I want to solve this math problem, and I need funding, research environment, collaborators, and access to mathematicians for discussions. All this I believe is provided by a PhD position in a University that is serious about path breaking research. My goal is to solve this particular problem and not any other problem.

How can I go about achieving this goal?

Edit : summary of problem

In effect i have proposed a new formula for inverse fourier transform with no assumption on its jumps except that they are finite and absent beyond a certain point. This new formula ensures not only pointwise convergence but also convegence of variation. Classical Fourier formula does not have convegence of variation in the presence of jumps.

Answer 1

Unfortunately, I expect it will be difficult to find a supervisor who will take you on under these circumstances. It's not impossible, but it will require finding exactly the right match, and that will require a lot of luck.

Here are some of the issues that will worry potential advisors:

1. What if the problem is too easy, and turns out not to be substantial enough for a Ph.D. thesis? Will you drop out from graduate school in disgust? Will you angrily insist that you signed up to write your dissertation on this topic and now they have to let you graduate?

2. What if the problem is too hard, and you spend several years in graduate school with no hint of progress or even the potential for future progress? When your advisor suggests changing course, will you listen? Or will you keep banging your head against this problem until they kick you out for lack of progress?

3. What if the problem doesn't end up being nearly as interesting or important as you hope? Will you accept this outcome, or will you turn into a crackpot who insists that the rest of the research community is just wrong?

4. Are you too emotionally invested in this problem to judge whether you are making progress or how interesting it is? How painful or awkward would it be to work with you on this topic?

5. Are you interested in learning how to do anything else besides work on this specific problem? The purpose of a Ph.D. program is to train researchers with some breadth, not just to solve one specific problem, so you need to maintain some perspective and longer-term goals than any single problem.

It's possible that none of these issues is an actual problem, but few people will agree to supervise this thesis unless they have enough knowledge of both you and your problem to feel reassured. Unfortunately, the bar for "enough knowledge" is probably pretty high. I wouldn't do it unless I knew you relatively well and had spent some time thinking seriously about the problem. This is not going to happen during the ordinary graduate admissions process.

Instead, I think you're probably best off not worrying too much about graduate admissions, and instead just focusing on convincing people that this is an important problem. If you can make a real intellectual connection with a professor based on this problem, then you can explore whether it might be a suitable topic for a Ph.D. thesis. Otherwise, applying to graduate school for the sole purpose of working on this problem is likely to be fruitless.

(To be clear, I might be wrong about this, but it's my impression based on the limited information in your question.)

Answer 2

It sounds like you have a singular focus on achieving the following two goals:

1. Getting into a fully funded, high quality pure math PhD program.

2. Using your position as a funded PhD student to solve a particular math problem you have formulated, and nothing else.

Let's start by thinking how you can achieve the first of these two goals. The thing to keep in mind is that PhD programs decide whom to admit based on the applicant's potential to successfully complete a PhD. So, the extent to which your plan formulated in item 2 above will help you get into the PhD program depends precisely on what it says about your potential to complete the PhD.

Unfortunately, by stating that you are uninterested in working on any other math problem, I think you are likely to hinder your application, and make it less likely that you are admitted, by showing that you have less potential to successfully complete a PhD than other students with more normal plans. The reason for this is that you are taking an approach to math research that any experienced mathematician knows is a bad one, which is to "put all your eggs in one basket". The fact is, most research problems that mathematicians work on end up leading nowhere, and for this reason experienced mathematicians know that in research one must be open minded and try to work on many problems to have a good chance of success, and that even when working on a specific problem one must be flexible and willing to change one's goal as one acquires a more nuanced understanding of the problem.

Likewise, if you dedicate yourself to solving your problem and doing nothing else, it is overwhelmingly more likely that one of the following things will happen rather than your dream scenario of solving it and getting a PhD based on that solution:

• The problem will end up being too difficult and you will not succeed in solving it.

• You will discover that the problem is known and is solved in some obscure paper from the 1950s, or is an easy corollary of a well-known result.

• You will discover that the problem is easy and has a trivial solution, and moreover the application you had in mind will turn out to be not as useful as you thought, making the result unpublishable.

• The problem will end up being at a good difficulty level. You will solve it after a year or two of work, only to discover that for subtle reasons you did not appreciate beforehand, the application you had in mind is not as useful as you thought, making the result publishable but not very interesting, and not enough to get a PhD for or to make you feel that the whole endeavor was worth the effort you put into it.

To summarize, your attitude as it currently stands positions you as a difficult person with unrealistic goals and expectations about what it means to be in a PhD program -- not a good image for someone trying to get into such a program. If you really want to achieve goal number 1, let alone goal number 2, you need to think about how you can present yourself as a more attractive candidate. As Anonymous Mathematician explained, that means being more open minded and expressing curiosity about getting a broad exposure to many areas of math and many research problems, and not just showing a singular desire to perform one feat you have set your mind on and which no one else is yet convinced is either worth doing or feasible to perform.

You also need to get rid of the mindset (as expressed in your comment "You know how difficult it is to come up with a provable mathematics statement thats not trivial and absent in the literature?") that by formulating a new math problem you have done something amazing that shows some special level of talent. You may have done something amazing, and you may be really talented, and the fact that you came up with a reasonable-sounding open problem does say something mildly nice about you, but frankly, open problems are a dime a dozen (one can easily find hundreds of them on MathOverflow). You can mention the problem in your Statement of Purpose, and perhaps some people on admissions committees will be impressed by it, but your unhealthy focus on the problem at the moment is much more likely to hurt your application than it is to help it.

Answer 3

You want to work on a particular problem. You do not yet seem to have all the tools needed to do independent research, such as literature searches, paper writing and notation skills, and a mathematics academic network. If you had all those skills your mathematics question would not have needed so many edits and would have been accompanied by a survey paper showing the current and historical state of the most closely related research that exists. A PhD would indeed be a very good way of building those skills, but you are trying to jump straight to research on your problem, skipping the tools-building step.

Instead, you should use the a PhD program for what PhD programs are designed to do, to aid you in becoming a fully competent researcher in your area. For that purpose, the choice of problems is much less important than what you learn working on them. You can learn, for example, to write publishable papers by writing papers about something in the general area, in close collaboration with published researchers.

Late in or immediately after the PhD you should revisit your problem, trying to step back from it emotionally and evaluate it as you should have learned to evaluate problems during your studies. Do you think it is important? Interesting to any community with journals etc. in which you could publish? Unsolved? Solvable with reasonable effort? If the answer it all those questions is "yes", either look for a post doc that will let you do independent research or look for a reasonably easy industry job that will leave you enough time and energy to work on your problem, and go for it.

Answer 4

I disagree with the previous answers. A student who has a serious proposal for a research project is highly motivated and probably gifted and well educated. So when it comes to hiring this would be a big bonus. As long as the proposal is slightly related to my own interests, I can give useful advice. Different from experimental fields, in mathematics a Ph.D. student does not help her advisor, if you are really interested in something, you do it yourself. In fact, it is important to be not too interested in their problems, because otherwise you would constantly have to pretend that you are not ahead of the student.

However, looking at your webpage I have great doubts that this proposal is well suited for a Ph.D. It deals with a problem in a rather old-fashioned area, so it might have been answered 80 years ago without many people today knowing so. The convergence questions you raise should be doable within a few weeks. The problem of approximating functions without increasing the variation is also not new, in fact this is one of the big advantages of splines.

Answer 5

Successful research professors are those that can get grant money for a specific research project. They then spend this money on doctoral candidates and post docs who will work on their project. It's very unlikely that a faculty member is going to devote much of his time and money to someone else's research project and generally it would not be ethical for him to do so.

Your best hope, and it's a slim one, is to apply for a research grant yourself.

Answer 6

For the reasons mentioned by the others, I would recommend that you search for an appropriate open science portal, and start working on the problem yourself. (It's better you don't join a PhD program just to tackle this problem).

Be sure to try and explain why you think it is important, and possible applications for it, and so on. The better you communicate, the more chances of people actually getting on board, and providing you with their expertise (which you will need).

It needs to gather steam in the community, which might take some time (assuming it doesn't get shot down immediately). Once you have had a few people go over it and give you their take on it, you'll be in a better position to know what to do next.

Answer 7

1. You want to work on a particular mathematics problem. You think you have nailed down the scope of the problem, and you think it is applicable to a real world problem. You are highly motivated. You feel that to make meaningful progress with this problem, you need to receive funding so you can devote yourself full time to this project, and easy access to collaborators. Although I don't think you said this, I got the impression you also feel you would benefit from some targeted coursework.

2. You have the idea that a PhD would be the way to go about things.

Did I get that about right? If so:

How about you try to put together a self-study program? Figure out what you need to learn; find a mentor; take some classes; attend seminars. To resolve the funding issue: don't quit your day job.

It may be that as you are going along this path, you will find yourself gravitating towards a PhD program for broader reasons than you currently have. And then that would be a natural next step.

For now, though, it sounds like you have the drive and interest to keep you going without a department providing a particular structure for your endeavor. (As others have pointed out, a department's structure doesn't align perfectly with your goals.)

Best wishes for your studies and for your project!

Answer 8

In your shoes, I would apply to a "bunch" of PhD programs, and write about your desire to take a PhD program for the purpose of solving this problem in your Statement of Purpose. I wouldn't go much further than this, and ask for funding and research grants until I got admitted. The single-minded focus on the one problem will give people a clue about your level of interest.

You may be asked to work on a few "other things," but hopefully, your posture will land you in a program that will allow you to spend most of your time (80% or more) pursuing your field of interest.

In giving you this advice, I'm assuming that you want a PhD for this one purpose only, and have no other reason for seeking a degree. If there are "other" reasons, I would make solving this problem only "part" of my Statement of Purpose.