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I will be completing high school at the end of this academic year. I have a strong love for Maths, and I am considering becoming a pure mathematician or pure mathematics researcher when I'm older. I have a few questions about the academic world of Maths in general:

  1. I have discovered some formulae, or theorems if you prefer, about a quite specific topic in maths with apparently limited, if any, applications to any other areas of maths. I would like to write a paper about it and try to publish it in a journal, but I'm unsure if it's been discovered before, as it hasn't come up on any Internet searches I've conducted. I've also searched arXiv for it. My question is, is there some sort of archive or database that I could use to quickly check if something I'm investigating has been discovered before?

  2. What exactly do research mathematicians do? Do they simply think all day and write/type their thoughts and results and progress as their job?

  3. If I don't discover anything new as a maths researcher, would I still be employed or would I no longer be payed, ie is it a safe and steady job?

  4. I would really enjoy being a pure mathematician/pure mathematics researcher, but is it a good job in terms of salary? I don't think job satisfaction would be an issue.

  5. To be a research mathematician do I need to very good at maths or be spectacular at it?

Thank you for your advice, it is really appreciated.

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    These are all good and valid questions and I do think you would benefit from a local mathematician who you could have a talk to. Could you ask your teacher to put you in touch with a mathematician? (Or, if not, maybe even another researcher?) Or does your nearest university has any person/program for people who'd like to study maths? You might want to check it out. – user111388 Dec 13 '20 at 17:40
  • @user111388 thanks, I'll ask my teacher if he can put me in touch with anyone. I'm not sure if he'll be able to though; are there any mathematicians on this site? – A-Level Student Dec 13 '20 at 17:56
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    The career advice section on Terry Tao's blog might help with some of your questions: terrytao.wordpress.com/career-advice, and I encourage you to try and find the websites of other mathematicians (or even twitter profiles) for other opinions. My own thoughts: a career in academia is very unstable and permanent jobs are hard to come by (this can depend on your field somewhat). It's better to be aware of this from the beginning. To be a good researcher I'd argue that perserverance, hard work and a strong interest in your subject are far more important than talent. – astronat Dec 13 '20 at 18:17
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    Go to university and study math. Mathematicians are sought after and well payed in industry and banking and insurances and whatnot. Decide on your next career step when you have the first degree. Then you will also have met enough profs to decide if thats something for you. – Karl Dec 13 '20 at 19:20
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    We've had similar questions in the past, perhaps try searching this site for "publishing while a high school student" or similar. It's important to be aware that your result might not be as novel as you think it is. Math Overflow is possibly the right place to post proofs and solicit discussion, but check their question policy before posting. If you really do have a publishable result, you'll almost certainly need help writing the paper itself. I echo user111388's advice: talk to your maths teacher or email PhD students/postdocs/lecturers in maths at some nearby universities. – astronat Dec 13 '20 at 20:31
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Let me try to give you some perspective on mathematics and mathematicians.

First, there are lots of kinds of mathematics. Algebra and Geometry were studied early in human history and other fields split off from them. At first glance these two seem quite different, but now there is Algebraic Geometry, a field of its own. There is also Analysis which studies certain kinds of relationships and Abstract Algebra which studies certain kinds of operations on things. No, there is more to it than that, but it gives you an idea. Topology is something like Analysis in some ways and something like Geometry in others. So there are some overlaps.

But the differences between mathematical fields are big enough that it is possible, even likely, that you can have true insight into only one or two of the many fields. And it is insight that is required for mathematics.

Mathematicians study problems. Applied mathematicians study "real world" problems and apply mathematics to their solution. Pure mathematicians study problems in mathematics itself. They study what has been shown to be true over time, but try to imagine what might be true and then set out to explore whether it is true or not. This is where insight is required.

It is unlikely, but not impossible, that you have discovered something new. It is less likely, but still not impossible that what you have discovered is significant. It may have been noticed in the past and not explored deeply, for example, seeming obvious to someone who has studied it deeply.

But, one thing that mathematicians do a lot of is work with others and share ideas. They bounce "new" ideas off of colleagues and get feedback. So, for someone in your situation, teachers and professors are the people most likely to be helpful at this stage.

If you want to be a mathematics researcher then put yourself on a path to eventually earning a PhD from a good school. Most pure mathematicians are also academics, though there are exceptions. Even an amateur can be a researcher, but will probably need to build up a circle of contacts and collaborators over time in order to be effective.

In a certain sense mathematicians work literally all the time. There are no breaks. But you don't sit at a desk for most of it. Your mind will work whether it feels like "work" at all. I often go to sleep thinking about some sticky problem and wake up with the solution. The insight comes sometimes when you let your brain relax, rather than trying to force it through some complex logical argument. The crux just falls into place.

Working with professors, especially on a doctorate, will give you a sense about what problems are important as well as ideas about how to attack them. But it is the "imagining what might be true" that makes a mathematician. The proofs are hard work and require training and practice, but it is that insight that makes the difference.

Only a few of us are "exceptional" as the word implies, or spectacular. But we all work toward excellence. Do that and you should be fine.


As a young person, I suggest you study widely (and not just mathematics). There is plenty of time to specialize later. And if you want to do serious work in any field, get in the habit of carrying a notebook (or equivalent) so that you can quickly jot down any insights or ideas that pop into your head while you are doing other things.

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  • Thank you for the extensive answer, I appreciate it. Do you know of anything that would be helpful for my first question, apart from discussing it with my teachers? Also, I'm not exactly sure what benefit there'd be from discussing my formulae with my teachers; do you mean for proofreading and finding possible holes in my reasoning? – A-Level Student Dec 13 '20 at 21:44
  • The problem with your first question is simply that the mathematical literature is so broad. Discussing your ideas with a teacher is probably best as they have a wider view. They might be able to put you in contact with someone who is a specialist through their own contacts. – Buffy Dec 13 '20 at 21:47
  • Thanks for elaborating, I will try to discuss my ideas with my teachers as you suggest. If my teachers don't have any useful contacts for me what do you suggest I do? Do you think it'd be a good idea to approach a journal with my paper, and tell them my age, once I have completed it? – A-Level Student Dec 13 '20 at 22:03
  • Your age will be irrelevant to them. But there are one or two journals that publish student work. Mostly undergrads. Sorry I don't have names. – Buffy Dec 13 '20 at 22:26
  • I see. What do you use to check if something you've found is indeed completely original and hasn't been found before? – A-Level Student Dec 14 '20 at 17:54
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To answer part of your question about financial concerns - If you are passionate about mathematics and likewise good at it, you should not ever have a problem finding employment, although it need not be a strictly math field. The knowledge one gets when learning higher level mathematics is applicable in nearly all technical fields, however it is more than that. It's not just about the knowledge you get, but rather, the intuition you end up developing about the world and logic in general.

Mathematics at its most basic level is abstracted from the study of logical structure and axioms. While most people are "logical", a mathematician inevitably develops the skill of automatically thinking about the subtle differences of objects and their properties, at a far more rigorous level than a person in any other field. A mathematician learns concepts that nobody in any other technical field ever gets exposed to; concepts which are universally applicable to pretty much everything in existence.

One thing I came to learn that nobody really talks about is that learning mathematics to a high/abstract level inevitably makes it far easier to learn pretty much any other technical field. Not only will the mathematical concepts in less technical fields be far easier to comprehend, but the mathematician will be able to recognize abstract features of topics in those fields that someone who specializes in the field may have never even thought about.

In short, while mathematics can be challenging, it is by far the most rewarding and universally applicable field a person can really study in my opinion, although this usually isn't obvious to people when first introduced to things like algebra and calculus. Another way to think about this is to consider that much of higher level mathematics involves the study of very abstract concepts and structures - and then ask yourself: What is the point of "abstracting" anything? The entire philosophy of abstraction is about making things easier, simpler, better, and so on. Such philosophy of abstraction is what we owe our entire civilization to.

But again, I wouldn't suggest someone jumping into pure mathematics if it is not something they are truly passionate about. However you define "passion" in that regard is up to you - people are passionate about different topics for different personal reasons.

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The other answers are focusing on "should I go into mathematics as a career?", but no-one has answered your first question, about what to do with your research results. There are related questions (with answers) on this site:

Your particular question about "how do I know if my work is novel {without established connections to university-level mathematicians)?" has not previously been answered, as far as I can tell.

  • in addition to ArXiv, you should search for published papers (ArXiv is only for preprints, and only a subset of papers are posted as preprints before publication).

    • mathscinet is probably the best index for math papers, but most access is through universities; I don't see an easy way to get access as a high school student (you could check into the London Mathematical Society, but membership ordinarily starts with undergraduates and isn't cheap).
    • Google Scholar is second-best, but has the distinct advantage of being free (it will find papers published in journals as well as on preprint servers like ArXiv: if you have trouble getting access to the text of papers you find, you could take a look at this question ...
    • someone suggested MathOverflow; I don't know whether that site (or Mathematics Stack Exchange) is an appropriate venue for this kind of question, but you can browse the questions & answers and read the information on those sites about what kinds of questions are appropriate. (If you find answers there you might not need the rest of my answer below ...)
  • if you find papers that contain exactly your results, you're done (for now), although you can and should pat yourself on the back for discovering a non-trivial mathematical result

  • if you don't find your result already published, look for papers that are closely related to your topic, and note their authors/e-mails (a contact e-mail is always listed in a published paper; it's also typically easy to find academics' e-mail addresses via web search).

    Provided that your maths teachers have looked over your work and think it's good, I would say it would then be OK to write a polite letter to one or two academic mathematical researchers who you have identified as likely having the expertise to evaluate your results and ask them what they think. (Do be polite (proper salutation etc.), and keep your e-mail short; state your results briefly and attach a slightly longer description that they can read if they want.)

    It is probably a good idea to write your idea up as completely as you can, in a format as close to a publication as possible (e.g. in LaTeX); this will make it easier for researchers to read (and they're more likely to take it seriously in the first place), and is the first step toward publication (and good practice!) anyway.

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Some addition to the other answers, related to your questions although maybe going a bit off track as I think this may be of interest to you:

(1) There are probably more jobs in which there is some math research besides other things, particularly in academia, like professors, university assistents etc. These involve math teaching, examining, and also administrative tasks. Even people whose job is to be a math researcher in the first place will have to do administrative tasks, apply for research money, write project suggestions and sit on the occasional committee. Also some companies employ researchers who then often work on projects, in groups, that are defined by the companies. Very often this is interdisciplinary and the mathematician collaborates (and actually competes for the positions) with people from, for example, computer science or biostatistics.

(2) Studying mathematics opens many possible career paths for you, and it can't hurt to have an open minded attitude to possible opportunities. There is much work to be done in physics and engineering, artificial intelligence, statistics that requires mathematical competence and often also mathematical research. Quite a bit of the research can be connected to computer simulation. If you focus on being a "pure mathematics researcher", there may not be that broad an offering of positions later (many pure researcher positions are fixed term and people are expected to use them to qualify for a professorship or another position that comes with some additional duties), however the overall spectrum of jobs in which mathematics research is done is very wide, and you may even find out that you love some of the other options as much as pure mathematics research. (I love teaching and statistics for example, so I became a statistics professor - I do a lot of things other than doing proper mathematical research, but at times I do that as well.)

(3) One can easily say you need to be very good, as in pretty much every job that is sophisticated enough. However you have to find out for yourself whether you are up for it and studying and doing a PhD in math is a good way of doing that. The positive thing is that at the point at which you may decide that a pure mathematics researcher career is not for you for whatever reason, chances are that you have lots of other options, many of them related.

(4) Regarding the third question, the "pressure to discover something new" looks very different from "outside the system" than from being in it, at, say PhD stage or higher. At some point when you have a good overview of a field, you will usually see a number of things that can be done with pretty low risk, i.e., you can address certain question of which you know that you have the necessary tools to solve them and chances are that they are not done yet. There may be competition, but very often different people can get a slightly different spin on things so that what looks pretty similar to something else can still be sold as "new". Obviously this is not what most people imagine to be exciting research, and at a sufficiently high level things certainly don't work like this, however there are many researchers who do research that is useful and to some extent new, and can even be some fun, without saying that they discover big new things.

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  • Thank you for the answer, it's quite thought-provoking. I am just wondering, why is there suddenly activity on my question again now? I've had a couple of upvotes, one comment and one answer (yours) so far today/yesterday. – A-Level Student Jan 22 at 12:24

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