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I am still working towards an undergraduate degree in Mathematics, however I don't believe it's possible to think about research too early. I'm thinking about going in the direction of Pure Mathematics - possibly in the area of Number Theory and Logic.

I have edited the above title, as I think it reflects more the question I was trying to ask.

I was really wondering if anyone firstly has an answer to my above question, and also if anyone could recommend good places online where one could follow current research activity in the area of Pure Mathematics. Number Theory, Logic, and Foundations of Mathematics particularly interest me.

Thanks in advance.

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    Why would you do this outside a university? You say you are pursuing an undergraduate degree. I recommend going about this within your university, and in particular by asking these questions to your professors. – Anonymous Mar 19 '14 at 23:30
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    Talk to your professors. Really, they don't bite. – JeffE Mar 20 '14 at 3:09
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    JeffE: If I had a professor to talk to then I would not be here. Unfortunately, due to being in a full-time job at the moment, pursuing research along the conventional route, may never be an option for me. So I'm having to think outside that box. (I would absolutely love to be doing research in a University if I could!) – Seraphina Mar 20 '14 at 10:36
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    @Seraphina: Can you clarify how you are "working towards an undergraduate degree in Mathematics", yet don't have a professor to talk to? – Mark Meckes Mar 20 '14 at 12:46
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    @Seraphina: Who teaches the classes you are taking? Even distance learning courses have instructors. To answer your question to Matthew: Some subfields of CS (like mine) are indistinguishable from mathematics. – JeffE Mar 20 '14 at 13:42
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I have published a little work in number theory while working full time. There is no big secret, just read about the problems that are at a level you can solve, solve the problems and publish the results.

Publishing in a journal as an unaffiliated researcher is harder than for professional academics but not impossible. Learning TeX will help because many journals want LaTex submissions. Most background information is available online if you search for it. You can join polymath projects, use mathoverflow etc

There are some advantages of working outside the system. You dont have to go around pleasing the people who might offer positions and you can work on what you want rather than what a math department or funding agency wants. So enjoy it as a hobby while earning a proper salary.

Update: I will add a little more specific advice. If you were doing a doctorate in a university you would have a wise old supervisor who would make sure you are heading in the right direction. If you want to work outside academia there is a danger of getting on a track that has no hope of success. Do not fall into the trap of thinking you can tackle a well known unsolved problem directly. This is a mistake very frequently made by outsiders.

It is better to start on simpler problems that you can handle and use them to build up expertise even if they are not of much interest to others. One way to find such problems is to start from a well-known one but then generalise it in some novel way to a bigger class of problems. Then look for special cases that can be solved and then generalise your solutions to solve a wider class of cases.

Finally, use your skills as a programmer to find solutions of interesting problems and then look for patterns. When you find patterns, make conjectures from them that you may then be able to prove. Try solving the problems at https://projecteuler.net/ to perfect your algorithm skills

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A substantial fraction of activity is visible on arXiv, the main page for which is at http://front.math.ucdavis.edu/

For example, the "number theory" listings are at http://front.math.ucdavis.edu/math.NT

But, also, many mathematicians keep papers on their web-pages, so you can look backward in time effectively on these pages. And there are listings of many number-theorists' URLs at http://www.numbertheory.org/

That is, with the internet, it is possible to keep in touch with current and past activity to a great degree. Many journal articles are behind paywalls, unfortunately, but substantial information about such material is available by other means.

...and: https://mathoverflow.net/ is a Q-and-A for research-level mathematics and mathematicians, while https://math.stackexchange.com/ is for undergrad and beginning graduate-level mathematics. Old questions and answers are searchable in both cases. In the case of math stack exchange, some dubious or not-ideal answers propagate, so take things "with a grain of salt".

But if you can find any way to have more direct personal contact with mathematicians, things can be communicated that you won't see in formal writing...

  • For tracking arXiv I would recommend scirate.com (feed + votes + comments). – Piotr Migdal Mar 20 '14 at 16:04
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I currently have the opportunity to spend a lot of time (meaning: not even a part-time job) thinking of a wealth of problems. This is great if I don't care about how I am going to handle my retirement (options: go homeless; die early; get adopted). I can fuss with details, spend time on the Internet researching or messing around, and still get enough sleep, time with family, etc.

You can lead your life as you choose. If you want to lead it best however, make sure that what you do is in line with your values. If you can get satisfaction from e.g. spending an hour a day thinking and writing math, schedule that in. If you absolutely want to integrate math into your current lifestyle, the major resource involved is time. Make sure that resource is spent wisely. (I've chosen to spend time away from math learning piano. We'll see in a decade or so how good a choice that was for me.)

Under the assumption that you have approximately ten hours a week to devote to mathematics, I suggest building Internet and personal resources. Involving yourself on online fora such as math.stackexchange and MathOverflow may help, if you don't clash too much with the culture. Write up your observations in ways that might be understandable by others. Engage with other people, and establish relationships (face-to-face or long distance) with the understood purpose of mutual mathematics enhancement. If you can get a mentor, even better. Find out if you like to solve problems ("repair work"), build bodies of knowledge ("catherdral or bazaar builder"), or note interesting connections (synthesis). If you can leverage your existing skills to the process, so much the better. Get to know your research library. If potential mentors have offered up their email, use it; don't be afraid to ask, and always respect the others' time.

Above all, manage your resources, primarily time, energy, and satisfaction level. Hopefully nature will tell you which direction to follow, and a support group will tell you (what not to do) to get there.

  • Thanks so much for all your really invaluable answers. I would like to mark them all up, but I can't do that at the moment. – Seraphina Mar 21 '14 at 15:12
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I would like to commend your mindset! This is exactly what makes mathematics more fun is the ability to go out and do it on your own without having someone else forcing you. The only problem it can be very ambiguous where to start.

I would recommend finding a challenging book and reading through it. Good subjects are in any university catalog (Calculus, DiffEQ, Prob, Number Theory, Algebra, Topology, Real Analysis, etc). Wolfram provides great definitions. Overleaf is a great place to publish documents. Lastly do at least a problem a workday!

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