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Sorry if this question is not well-suited here, but I thought research in mathematics can be identified from other science field, so I wanted to ask to mathematicians.

I am just starting graduate study in mathematics (and my bachelor was in other field) so I have no research experience in mathematics. Recently I came up with a problem by myself, and thought it was interesting so devoted some time to draw the results. Now I have some results, but I am not sure this is already studied somewhere by someone. I tried to lookup some possible keywords in Google Scholar, but have nothing.

I am sure for more mature mathematicians they already know their fields of study and recent trends of the research, so it won't be a difficult problem to know this is original or not. But if you come up with some idea that are not seems to belong to any field, how do you know your result is original or not trivial result?

Thank you in advance!

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    $\begingroup$ Since you are starting a graduate study, I would recommend you to choose an advisor and to ask him/her, whether your problem is interesting and whether your results are original. $\endgroup$ – Mikhail Borovoi May 21 at 15:32
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    $\begingroup$ The first question is whether your problem is interesting. A (good) professional knows what is interesting. Here the advisor can help. $\endgroup$ – Mikhail Borovoi May 21 at 15:43
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    $\begingroup$ Usually you know the history of your subject well enough to understand up to what point your problem is interesting and/or new, but there remains the possibility that the same problem has been tackled in a different form and different terminology in another area. It is still interesting to make the connection even if it takes away from your claims to originality. $\endgroup$ – alvarezpaiva May 21 at 15:45
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    $\begingroup$ @JingeonAn "do you think this kind of situations (lack of knowledge on the standard terms) won't happen for professional mathematicians?": in my case, yes, frequently. Sometimes converging to the right keywords takes time. In one case I was looking for reference to a result which sounded "too basic to be unknown", I spent hours without success, tried again weeks later with new ideas of keywords, found something which led me to try to ask some researcher (which I don't know, not in my department) and he confirmed me that it's standard, pointed out the right refs, the used terminology... $\endgroup$ – YCor May 21 at 16:12
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    $\begingroup$ *"do you think this kind of situations (lack of knowledge on the standard terms) won't happen for professional mathematicians?" In my case, yes. Once I learned about previous work on the topic of my paper from the referee report. Then I had to revise the paper significantly, in particular, to change the title. $\endgroup$ – Mikhail Borovoi May 21 at 16:17
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(1) It depends a lot on the field. In fields that rely on specialized techniques discovered relatively recently or known only to a few, or fields where the questions involve recently-introduced objects, it's much easier to keep abreast of current research.

On the other hand, in fields with elementary questions that could have been studied a hundred years ago, sometimes even senior mathematicians discover that their work was studied a hundred years ago.

Of course, working in a trendy field carries its own risk, that someone else could be working on the same thing at the same time, but not much can be done about that.

(2) If you're working in a specialized field, as other have said, the best thing is to ask your advisor. If you have an advisor in a specialized field and have ideas in a different field, the best thing would be to ask someone in that field. As a grad student you probably want to start with fellow grad students, but a senior mathematician would probably asks someone on their own level.

If you have an idea that is more elementary, you should still ask your advisor, but there are certain mathematicians who know a lot of elementary and classical mathematics you could potentially ask.

(3) With regards to literature review, one trick that helps a bit when keyword searches fail is to use citations. If your idea generalizes work of Paper X, or answers a question from Paper X, or uses in a fundamental way the results of Paper X, anyone else who had the same idea would likely cite the same paper. You can produce a list of papers citing Paper X on both Google and MathSciNet.

(4) As a starting graduate student, even if your idea is completely new and original, it is likely that the greatest value it provides to you will be as practice for your future work. (I mean if you're good enough to do groundbreaking work right off the bat, you will probably do even more groundbreaking work once you get some experience under your belt.)

So don't feel bad at all if you find out something was already well-known - the experience of formulating and solving your own problem makes you well-placed to do original research once you learn a bit more, as compared to someone who knows a lot but hasn't done this.

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    $\begingroup$ A lot of my work has been contemporaneous, or nearly so, with other mathematicians. However, there is value in adding a new perspective or application. Don't worry about originality. Do worry about doing a literature search. You aren't expected to find the hundred year old paper quickly, but you are expected to keep looking. You are also expected to familiarize yourself with the basics. Gerhard "Advisor Shouldn't Do Your Job" Paseman, 2020.05.21. $\endgroup$ – Gerhard Paseman May 21 at 16:27
  • $\begingroup$ Good advises, thank you! $\endgroup$ – Jingeon An May 21 at 17:56
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As was echoed in the comments by YCor and Mikhail Borovoi, the question of originality (especially for results that could have been stated a long time ago) is one that is relevant to all mathematicians. An interesting recent example that comes to mind is this story on Terence Tao's blog.

So how can you guard yourself from this? I think that there are two tools you can use:

  1. Experience. When you're around in an area for a few years, you should develop a pretty good sense of the literature. This is where an advisor can be really helpful when you're young, because by definition you don't have much experience at that point.

    Experience also means knowing the correct search tools, like looking in papers that cite a given one. But sometimes you just don't know the correct words to search for, so then there is:

  2. Asking other people. This is related to 1: one of the most useful skills is knowing whom to ask, and this again comes from experience and connections developed over a long period of time.

In both of these your advisor has an advantage because they've been around longer. But as you go on you will find that this becomes easier to do yourself.

Of course research output is not the only type of mathematical productivity, and you can still learn a lot by rediscovering an existing result. But it's incredibly frustrating to find out something you proved is already known (especially when you don't have so many papers yet), because it changes the meaning of the time investment you made.

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If you just started your graduate study, you probably have an adviser. If you don't have one yet, try to find one as soon as possible. Anyway, in most universities that I now an adviser is required to defend a PhD.

Then show your result to your adviser. Even if it is from a different area from his/her scientific interests. Adviser will probably make a judgement, or if necessary will know whom to ask and where to look.

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It's not that hard nowadays to do a fairly exhaustive literature search to make sure something hasn't been done before. This is a skill which can take a bit of time to develop but will help and saves a lot of time in the long run, especially if you do not have someone you can show the result to. This will also help you to be independent, something which is starting to become increasingly important in academia (especially in mathematics).

Now with Google, Arxiv and many other tools which people in the past did not have access to it is much easier to check if something has already been done, although these checks are usually not foolproof. After a while you will probably also develop intuition for what is already a 'reasonably obvious' idea.

For example, a few weeks ago it occurred me that it would be nice to construct a quantum mechanical system whose Berry phase was a BPS t'Hooft-Polyakov monopole. It wasn't too hard to do some digging around and find that this had already been done: I encourage you to learn how to do this effectively if you do not already know how.

Edit: See this paper by Tao et al. where they investigate in great detail whether or not a result they have found in linear algebra is new or not and if not, exactly what is the history of the discovery. In Figure 1, they provide a very detailed tree of all the citations and previous mentions/studies of the result. Tao does not work directly in the field of linear algebra and his 3 co-authors are neutrino physicists, so how could they possibly have found this out except by doing some digging and deep searching around? OK, the graph might have been crowdsourced, but I doubt everyone contributing to it was exactly an expert.

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  • $\begingroup$ Yes, I am willing to develop the skill, and I agree it is not as hard as before. Then this may be a trivial question : how do you develop the skill? I guess the answer is by doing it. Maybe I just want some tips from more experienced.. $\endgroup$ – Jingeon An May 22 at 1:30
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    $\begingroup$ It's not that hard nowadays to do a fairly exhaustive literature search to make sure something hasn't been done before. - I often find it very hard to determine whether something I'm wondering about (in my field or not) is known. A big issue is that the way you formulate something often won't match the way other people do. Also, papers can be vague/unclear. $\endgroup$ – Kimball May 22 at 2:15
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    $\begingroup$ It’s not easy to do an exhaustive literature search — especially if you don’t read Chinese. $\endgroup$ – Matt F. May 22 at 3:36
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    $\begingroup$ I don't really agree with this. This scenario is common: you come up with an idea in connection with problem A, it has previously been studied in connection with problem B, the terminology used is natural if you are thinking about B but you would never guess it if thinking about A. $\endgroup$ – Neil Strickland May 22 at 8:59
  • $\begingroup$ I'm not saying it's foolproof to do a literature search, but it can be extremely helpful. $\endgroup$ – Hollis Williams May 22 at 12:29

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