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I'm not sure that this is the right exchange for this question. It asks about the possibilities of research in mathematics and computer science.

Extended Background

I am very interested in mathematics and theoretical computer science. Over the past few years I've gained lots of experience from math competitions, programming competitions, software development internships, and talking to professors, and I've learned that I do not enjoy writing code; I enjoy writing "beautiful" and elegant code in practice, just as I enjoy beautiful and elegant proofs in mathematics, but I do not enjoy writing hundreds of trivial imperative lines of code to make some company money. I've learned that I like to think critically about problems, and I like finding the quickest, most efficient, most elegant solutions. I enjoy the mathematical (or theoretical) side of computer science.

So, recently, after coding only with imperative languages my whole life, I discovered Haskell, a functional programming language. It is extremely close to mathematics: Everything is a function. Ideas are defined rather than executed; instead of giving a computer step-by-step instructions, the computer is given a definition. This discovery reaffirmed my passion for mathematics and the mathematical and theoretical side of computer science.

Minimal Background/Question

I've also been frequenting a website called Project Euler, which is filled with 450+ mathematics-based programming problems. This is essentially the epitome of my passion. I love solving these problems with Haskell (and sometimes, when I'm clever enough, with merely a pen and paper).

I know that many of the questions (if not all of them) have already been solved and researched in both the field of mathematics and the field of computer science, and hundreds of their variants have been as well. However, many of these problems were initially proposed and solved hundreds of years ago by mathematicians (such as Euler and Gauss), and most of the relatively newer problems were all proposed and solved by the 1980s.

So my question is, are there still problems like this that I can think about for a living? If so, which field and sub-field is closest to this type of research? If not, would I be involved in these problems by teaching and then introducing them to students of mine, or hosting competitions?

TL;DR

  • Can I solve problems like this one as a career?
  • If this research isn't viable, would I be more involved with this type of problems by teaching and hosting competitions, etc?
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    I assume by research you mean paid research rather than just a hobby. If so, this probably makes sense mainly in an academic environment. So it would be useful for us to know whether you have any relevant academic background. – Bitwise Aug 26 '14 at 18:40
  • @Bitwise I am still only a senior in high school, though I have been actively involved in mathematics and computer science for quite some time now. I have explored my passion for academia over the past few years and have firmly decided that I want to be involved in it for the rest of my life. – user20284 Aug 26 '14 at 18:42
  • Have you considered doing research in theoretical computer science, for instance on programming languages? – Federico Poloni Aug 27 '14 at 8:20
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    Most code isn't trivial. If you are spending a significant fraction of your time writing trivial code, something has gone wrong. Automate it. – Patrick Collins Aug 27 '14 at 9:14
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    Your enthusiasm will serve you well in any career that you choose, but I would caution you that it is increasingly difficult to pursue a career as a tenure-track mathematician. There are many software companies that solve difficult, math-intense problems; it's not all writing line-of-business software for medical records automation. If you're interested in learning about the sorts of problems that professional mathematicians solve, go to your local university library and take a look at some journals; the librarians will help you find what you're looking for. – Eric Lippert Aug 27 '14 at 13:42
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This may be a better question for Math.SE. That said...

These sorts of problems fall in the category of contest math. Sometimes they are related to active areas of mathematical research (most often number theory, combinatorics, and geometry), but more often, current research deals with problems that are considerably more complex and can't be solved (or even stated) in a page or two.

Note also that, at least in the US, there are very few people who get paid to do mathematics research full-time and exclusively. Most professional mathematics researchers are professors at colleges and universities, and their duties also include teaching and administration (to varying degrees).

There is a recognized subculture of mathematicians interested in contest math. They may get involved in creating problems, organizing contests, and coaching students. At some universities, such activities may be considered a significant part of their research or "scholarship" duties, but they would normally be teaching regular math classes as well.

Edit: Actually, there is a category of careers I had forgotten: intelligence. The NSA and its counterparts (GCHQ, etc) employ many thousands of mathematicians. Of course, it's hard for an outsider to know what goes on there, but it could be that their activities have more of a problem-solving flavor. At least they are not bound by the requirement that their work be publishable! The NSA has a well-known (and highly competitive) summer internship program for undergrads, the Director's Summer Program, so that could be one way of testing those waters relatively early on.

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    Well put. <characters> – paul garrett Aug 26 '14 at 22:26
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    "in the US, there are very few people who get paid to do mathematics research full-time." But I think it's worth noting that there are a substantial number of people who get paid to do mathematics research for a substantial fraction of their time. – David Richerby Aug 27 '14 at 7:51
  • @David: Well sure, I happen to be one of them. :-) The asker seemed interested in doing math more or less exclusively, by my reading, so I wanted to address that. Though I just thought of something to add. – Nate Eldredge Aug 27 '14 at 12:50
  • @NateEldredge I know you know that but I also know the asker doesn't so I think the clarification is helpful. – David Richerby Aug 27 '14 at 13:01
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Admins please forgive me (and inform me so I can delete this) if this is too off-topic from the question.

I do not enjoy writing hundreds of trivial imperative lines of code to make some company money.

Where I am now: I'm in my mid-twenties, somewhat recently out of college with a B.S. in CS. I work for a company that makes business software, (making them money, of course).

Where I was a few years ago or so: I got into a top-20 school (U.S.) and initially went there going for a B.A. in Math, fully intending on spending the rest of my life solving theoretical math problems and going on to get a PhD (that was actually my dream since middle school or early high school - can't remember exactly when). Your sentiment reminded me of my own (which is why I'm writing this right now). Well eventually I changed my mind, but to make a long story short - I'm glad I did - I make more money (as a software engineer than an entry PhD professor at a local public university according to the public salaries) and what I do now is insanely fun, definitely more so than I would have thought. (Also walking into a room full of hundreds of people using something you helped create to make their lives easier is a cool feeling.)

I've learned that I like to think critically about problems, and I like finding the quickest, most efficient, most elegant solutions.

When you're making a business money there is still plenty of opportunity to think critically about problems (at least where I work), and you can find still find quick, efficient and elegant solutions (I somewhat shudder about superlative evaluations of software engineering solutions - it's all trade offs and compromise in your solutions, which corner of the CAP theorem do you sacrifice in exchange for the other two, and why, etc.) to the problems; it's just one of your implicit constraints is money.

My advice: Do what you enjoy, of course, and don't let others impose their dreams for you on you, be it your parents, friends or strangers on the internet, but do keep an open mind on what it is you do with your life. (And please excuse all the parenthesis :) )

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    It might not answer the question but it is a very good advice - the OP is still very young and the best thing to do is to be open minded. Things are not always what they seem to be, I can state that from experience, moving to the industry after nearly 30 years in academia. As strange as it may sound, I have more time for research now. And the results of my research are going to be used straight away, which is often not the case in academia. – greenfingers Aug 27 '14 at 8:29
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    @greenfingers: I find it very patronising to answer the question "I don't want to do X, I want to do Y, is it possible?" by "Actually, you could enjoy doing X". I don't think the OP asked for a list of positions where critical thinking is required. It might be a piece of advice, but it is not an answer to the question. – user102 Aug 27 '14 at 8:36
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    @CharlesMorisset - "Do what you enjoy, of course, ... but do keep an open mind" - don't see anything patronising about this – greenfingers Aug 27 '14 at 11:37
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    "I make more money (as a software engineer than an entry PhD professor...)": For some people, money doesn't define wealth. – jdoicj Aug 27 '14 at 12:12
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    @Charles: The OP is technically a child, so "patronization" is less inappropriate. Of course it is still a matter of degree, but...do you remember what high school was like? I sincerely doubt that a high school student's reaction to this answer and greenfingers comment would be "Where does he get off telling me what to do?" – Pete L. Clark Aug 27 '14 at 16:02
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I believe @Mangara's comment is very relevant here:

Number theory, combinatorics and geometry (the areas Project Euler mainly draws from) actually have lots of accessible problems that are still unresolved.

Accessible here means that the problem can be easily explained. However, unresolved problems that anyone can understand tend to be extremely difficult. If they weren't, then someone would have solved them!

Get a copy of Richard Guy's Unsolved Problems in Number Theory. There you can find some problems that constitute "real mathematical research" but are more or less in the vein you describe.

You can usually only (get paid to) do this research in a university, you have to be very talented to get a tenured job in these areas, and it's a long road. Also, due to the highly interconnected nature of mathematics, the techniques used to solve these problems are often vastly different from what you might expect -- for a famous example, see the proof of Fermat's Last Theorem via algebraic curves. So if you want to work in (say) number theory, you'll still need to learn deeply about and use other areas of mathematics.

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    I prefer his Unsolved Problems in Geometry. – JeffE Aug 27 '14 at 11:55
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As @NateEldredge already explained, such easily-apprisable questions, most often arising in math contests, or as puzzles, are mostly not what could allow a person to make a living as an academic mathematician, except in cases of ultimate-extreme talent, perhaps.

At least as idealized, academic mathematics aims to develop and validate viewpoints that clarify fundamental issues. "Problem solving" is significant, but is at the extreme end of the phenomenological aspects of the business. As Nate E. noted, most "active" questions in mathematics will not be easily understood by amateurs, or perhaps even by non-specialists. These questions are mostly not of the easily-apprisable sort...

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    Number theory, combinatorics and geometry (the areas Project Euler mainly draws from) actually have lots of accessible problems that are still unresolved. – Mangara Aug 27 '14 at 0:27
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    @Mangara, I have some reservations about "accessible problems that are still unresolved". That this is usefully so is a pleasant optimism, but I doubt that it is matched "in the fact". – paul garrett Aug 27 '14 at 1:25
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    I can only really speak for "the fact" my own field (computational geometry), but if we define an accessible problem as one whose problem statement can be easily understood by amateur mathematicians, many of the problems we study are indeed accessible. – Mangara Aug 27 '14 at 2:21
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    What @Mangara said. It is constantly surprising to me just how many accessible questions are open. The fact that a question is easy to understand is no indication that it is easy to solve! – JeffE Aug 27 '14 at 11:54
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    @Mangara: Still, even if the problems look similar to contest problems, the experience of solving them is very different. – Nate Eldredge Aug 27 '14 at 17:41
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Here's the view from my vantage point:

  • The kind of interesting problems you can solve in 30 minutes to 7 days each (as found in mathematics olympiads, competitive programming contests, and which, by the way, have tremendous value by being fun for whoever gets into them and being accessible to large numbers of high-school students, as well as developing their mental capacities and practical skills), are typically not of interest to most professional mathematicians because they are considered too trivial. Research publications require more substance than a couple of observations and a direct application of some 100-year-old theorem. Also, you would drown in the task of citing the relevant literature.

  • Therefore, as a professional research mathematician, if you want to keep your job, you will be effectively forced to find a frontier somewhere in some field and publish results which are very likely to be novel and well-received by journals. You can expect to meet lots of interesting sub-problems, but also lots of questions that frustratingly don't have easy answers. Also, the chances of this type of work making impact on some field external to mathematics are not great.

  • Interesting sub-problems come up all the time in all sorts of other research, not just in mathematics. Even in programming jobs that are not considered research — sometimes a requirement says that certain aspects have to behave in a certain way and meet certain constraints, and naïve solutions turn out to not be good enough.

Putting it all together

To alter the question a bit, I think there is a wide selection of jobs where 10-30% of your time will be spent on solving interesting sub-problems. These are not restricted to research in mathematics. Very few jobs where this figure will be 100% (people do this on a volunteering basis, verifying problems for Codeforces, TopCoder, etc., not to mention actually competing just for the fun of it).

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