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I'm 22 years old and studied 2 years of math before droping out to travel the world as an assistant of a friend who make commercials, documentaries and media creation in general.

I really love math and cs however I love to travel and learn about new cultures, so I can't go back to school in the near future, I'm getting more responsibilities and probably continue doing something similar for living like photography or videography.

I was wondering is possible to become an amateur mathematician doing relevant research independently? I can study 4 hr everyday, no more because of my work and because I have many hobbies. Also I can't work intellectually in a subject for more than 4 hrs. (I use to practice shogi and learn that 4 hrs distributed in a day work best for me).

I've liked algorithms since I'm a child, When I was in elementary school I liked this girl but I was super shy so I thought that if everybody in the class submit an ordered list of the people who liked we could make an algorithm to match people. I develop an algorithm who would make this possible and I remember that not was that simple, almost took me a whole year.

I think I have strong basics in abstract algebra, real analysis and linear algebra. I also have self study algorithms, computer complexity, basic computer theory and basic computational geometry. But now I don't know how to keep going. I also know to program because I enjoy making programs to solve problems like python scripts to automate download series or to automate the things I repeat a lot. For the things I learn I probably could get a CS degree since I'm interested in this themes since I was 12 and took one robotics course, I didn't learn much robotics but I was amazed with computers so I started programming.

I'm not interested in money, positions in universities or even social prestige (I probably would use TrashPanda as my penName like the boubarky school of math). Because I think academia has some big problems like the way academical publishing works. But as you probably has noted I really like fake internet prestige and I think don't would be cool that someone with no formal math education and with a pen name like trash panda started to publish in academic publications.

I admire profundy Erick Demaine and I after "reading"(I couldn't comprenhend many of the paper he have) some of his papers and the page in the CSAIL of MIT I think I'm interested in:

  • Discrete and computational geometry: Folding and unfolding, linkages, robotics, motion planning, dissections, simple polygonizations
  • Algorithms and their analysis: Adaptive computation, graph algorithms, string matching, randomized algorithms, approximation algorithms, fixed-parameter algorithms, streaming algorithms
  • Combinatorics: Discrete mathematics, graph theory (matchings, minors, treewidth, …), combinatorial game theory .

My plan is to self learn combinatorics (With the books: Principles and Techniques in Combinatorics by Chen Chuan Chong and Brualdis book in combinatorics). Dominate the book CLRS of algorithm analysis. For CG I plan to complete Devadoss & O'Rourke's Discrete & Computational Geometry (I'm half way), the Springer published Computational Geometry: Algorithms and Applications, and David Mount's notes (Page on Umd) .

But I figurated that I can start reading some papers that are not that hard, I pass a lot of time in Mexico and in Chile so probably I can look from a professor there that could help me.

Or maybe someone here would like to mentor me. (I promise that I could research a lot here or in mathexchange before as you)

What you guys recommend? I feel like when you like a person so much but you can't see her/him.

Is there some papers I should read? How can I discover a proper theme acording my current level to try to investigate?

Thanks in advance to everybody and obligatory sfmbe.

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    $\begingroup$ $4$ hours a day usually isn't enough to make much progress, to be honest. $\endgroup$ – Don Thousand Feb 23 at 3:08
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    $\begingroup$ I don't want to be discouraging, but it can be difficult to pursue "serious" research outside of academia for a large number of reasons. Most notably, you don't have other people actively researching around you, so you're missing out on great conversations, which help solve problems. Plus, when research isn't (at least part of) your job, it can be really easy to put it off, much like practicing an instrument. $\endgroup$ – HallaSurvivor Feb 23 at 3:15
  • $\begingroup$ That said, there have been exceptions before, and if you're willing to put the time in, Combinatorics is a really rich field, with a lot of problems that are fairly simple to understand, which makes them fairly accessible for anyone with the time and the desire to think about them. If you can find a real-life mentor, or even a friend who is also willing to put the time in to work on some problems, it would help greatly. And remember, math can be fun for its own sake! There are hundreds of "amateur" mathematicians, and even if their work isn't game-changing, they still love the work they do ^_^ $\endgroup$ – HallaSurvivor Feb 23 at 3:18
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    $\begingroup$ @HallaSurvivor completely agree. Also, I think that doing research in fields like CS is a lot more doable part-time, and can be math focused. (Also, you made me remember how I've forgotten to play violin in 3 years. Yikes) $\endgroup$ – Don Thousand Feb 23 at 3:19
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    $\begingroup$ The mathematician Hardy reported doing 4 hours of math per day (in A Mathematician's Apology). The book The Mathematician's Survival Guide says that if you can do 5-6 hours of focused work, 5-6 days a week, then you're doing good. I think 4 hours per day is fine. Just go deep in an area you're interested in. (As has been mentioned, your progress will be much faster and probability of success higher if you happen to have friends who are great at math and you can absorb their knowledge.) $\endgroup$ – littleO Feb 23 at 4:11
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What you want to do is not necessarily impossible, but it is unlikely, and harder than you think.

First of all, you seem to underestimate what it takes to become a researcher. There is no uniform rule, as there are wildly different levels of talent, and also different areas require different levels of expertise to be able to produce research. That said, a mathematician typically requires (in North America, but things are mostly equivalent elsewhere), to be at the stage of being an independent researcher,

  • A 4-year B.Sc. honours, followed by

  • a 2-year M.Sc., followed by

  • a 4-year Ph.D., followed by

  • One or two postdocs, maybe 2 years each.

From what you mention, you sound like you are at about half-way through the first step. During most of the above-mentioned periods, a normal person would be devoting more than 4 hours per day to their math.

Besides the great amount of hours required, these are not just "study time". There is a whole maturation process, where new notions and ideas become natural and one can use them with easy at the next level. Eventually, exposure to research comes in the form of reading papers, but also attending seminars and conferences, receiving feedback on one's own talks and classes, and one-on-one talk with other fellow mathematicians. Avoiding all of these would make it a lot harder to get to be thinking math at a research level. And, more importantly, it would be really hard to get to know what problems the community is interested in.

In summary, you can study as much math as you want on your own, but if you care for whatever you do to be appreciated by others, it is unlikely that it will happen without interaction with the math community. And this might be hard, even if you are working seriously: every mathematician is familiar with "cranks" who write at them or try to engage at conferences to convince them that their "research" is great. It (almost) never is, so there is a strong tendency to dismiss people form outside the academic world.

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  • $\begingroup$ Agree entirely but how can I get in touch with the math community. I travel almost all year and never stay more than 2 or 3 months in a place. Actually that was my question. Seems I explain it badly. What I mean as independet is that I don't work at a school. $\endgroup$ – Trash Panda Feb 23 at 4:30
  • $\begingroup$ I don't really have a solution for you, I was just explaining how things usually work. Over those many years of masters, doctorate, and postdoc, prospective mathematicians get to know many colleagues and later use those contacts in their careers. It may be possible afterwards to more or less have a career via remote communication, but it will be super hard to establish one to start with. $\endgroup$ – Martin Argerami Feb 23 at 10:35
  • $\begingroup$ Precisely math.stackexchange.com and mathoverflow.net are the places where you can begin to get the feeling of what is relevant and what is not. It is as good as it gets about interacting with mathematicians. Said this, I fully agree with @MartinArgerami, not impossible, but in practice highly unlikely $\endgroup$ – Euclean Feb 23 at 13:49

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