As a grad student how to approach a possible breakthrough research?

Question

I am in this peculiar position whereby I started off my grad school in subject X and got very boring projects in it and eventually around my 3rd year I started studying a new subject Y (partly because I met some inspiring profs in Y who motivated me)

Now in some sense it was just lucky that after about 6-7 months of working in this new field I along with another co-grad student hit upon a breakthrough research idea possibly solving a famous long-standing open question in this field Y! It was sheer luck that we hit upon an idea which seems to work! (though we are struggling hard to prove the theorem)

Now when I am trying to officially shift departments/institutes to pursue Y, I am explaining in my statements this new possible breakthrough that we hit upon. But I am a bit worried as to how senior established profs in Y will react seeing such a thing in my statement.

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I think all answers till now are completely missing the point of this question and what I am worried about. So let me break-up the context in parts.

(1) In mathematical stuff no one wants to put on arxiv a "possible" method unless one has a rigorous proof for it to work. If we had a proof then I wouldn't be asking this question!

(2) In my SOP if I don't write about this progress then what else do I write about!? This is the main stuff that I have been working on ever since I decided to shift fields. So I don't see a choice but to write about this idea!

(3) The professors who have seen the idea seem to have no opinion at all. They take the view that unless there is a rigorous proof how do we know whether this is right or not. At this point all evidence is numerical simulations. (and the prof who originally inspired me to start Y remarked that he had himself independently also tried one of the ideas we came up with)

(4) I have anyway linked from my SOP a draft which explains the idea.

Answer 1

As a good researcher always be cautious with "breakthroughs". I would explain that you think it is a good idea worth pursuing. Coming off too strong may put new professors off. Unfortunately I have witnessed many times people believe they have come across breakthroughs when they are nothing of the sort. I think partly because unlike how science is portrayed in the media, "most" excellent science is a long grind, through years of experiments, thought, collaboration and trial and error. The road is often long and winding, and to approach someone well versed in the field with a "breakthrough" may be hubris. But, if it is a strong idea and worth pursuing with good reasoning/logic/science behind it, I cant see a problem convincing anybody in your field the importance of what you propose! Good luck!

Answer 2

As I suggested in a comment, I think a key issue is whether your wish to switch to topic Y is conditional on the validity of your breakthrough. If it is, I agree with prior advice to document your priority on the idea and then start presenting it to professors as "Does this direction have any merit?".

If your wish to switch to topic Y is unconditional, there is an alternative strategy of delaying presenting your idea.

In order to have an opinion on the originality of your idea, you must have done a formidable amount of study of Y, including the history of attempts to solve the problem you are working on and related topics, despite the relatively short time you have been working on it. For purposes of an application SOP, you could present yourself as having become extremely interested in Y and wanting to do research in it.

That plan gives you a good chance of getting into Y research with no embarrassment if every serious, experienced Y researcher is already aware of flaws in your idea. It avoids the risk of being seen as a crank - the statistically most likely cause of an inexperienced researcher thinking they have a breakthrough on a long unsolved problem.

Answer 3

This is my advice,

You and your co-grad student write up your idea with your plan of attacking the solution and put it on arXiv. Then talk to the professors who motivated you to study subject Y to get their opinions. If the professors think you do have a possible breakthrough, ask them if you can shift to their department/school. If their answer is positive, start the formal application process.

I am not sure if you are worried that your idea could be stolen. This is why I suggest you to put it on arXiv first. Once it's on arXiv, that worry is gone. You should feel free to talk to the professors. If your idea is indeed workable, you have options.

You can stay in your current school/department so you won't waste your three years there. If you want to shift department or school, you'll have very strong recommendation letters from those professors because you already have possible breakthroughs.

You don't need to worry about the reactions from those professors. They are the experts in Y. They can tell if your idea is good after you present it. If it's not workable, you just lost six months of time and it's better you learn that fact before you waste even more time on it. If your idea is workable, they are going to be very happy because you have a breakthrough and you are their students.

Writing an SoP for the breakthrough at this moment may not be a good idea unless you are fairly certain that it is workable. Frankly, two graduate students spending 6 months to solve a famous long-standing open question is doubtful, but it is not unlikely. You do need experts to confirm it.