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I was wondering, what are the consequences of solving a hard mathematical problem or discovering something new?

I suppose the person would win some award if the achievement is good enough or money if there was a reward, but I´m more interested in knowing, for example, if universities will want those who solve problems to realize mathematical research in them or if they would want them to teach about the problem they just solve, would companies be interested in them? What jobs could they get? Would it be easy to them to find jobs with good salaries?

I realize that the questions asked in Math.stackexchange are more "mathematical", but I feel that those interested in answering math questions are usually also informed in the subject of my question.

Thanks!

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    $\begingroup$ You can publish a paper about it. A sufficient number of papers of a sufficiently high quality may make it possible to become a professor, and publications also look good on your resume when applying for industry positions. If it's a real breakthrough there could be an award but that is very rare. $\endgroup$ – littleO Aug 27 '16 at 23:47
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    $\begingroup$ Questions about what mathematicians do professionally and what it's like to be a mathematician are perfectly fine here (though some might be better suited for academia.stackexchange). $\endgroup$ – anomaly Aug 27 '16 at 23:57
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I'm not sure what exactly you're asking here. Of course universities want professors who have actually solved mathematical problems or discovered something new. Industry doesn't really care; it's mostly concerned with taking existing discoveries and finding applications for them. Even in the more ambitious or creative fields, you're not going to find available applications for most of mathematics, and even fewer for ones that people will actually pay for.

That having been said, it's easy to get a reasonably high-paying and prestigious job in industry as a generic newly-minted PhD, and it's trivial if you're a professor with decent bona fides. The main difficulties are finding something that you'd be interested in and switching from the academic mindset and environment (freedom to choose your own research topics, collaboration with colleagues, long-term or pure research) to the industry ones (management and hierarchies, demand for short-term results, and far less choice in what you pursue).

Since I don't know your background, I'll also point here that "solving problems" isn't exactly the goal of mathematics, at least for problems that are along the lines of Math Olympiad, etc. offerings. They're more like "Classify all groups that can act sufficiently nicely on $S^n$," or "Find whether there's an analogue of the Atiyah-Singer theorem for this new kind of structure," or "Determine whether all curves of this particular algebraic structure arise from this arithmetic structure." They're multi-year projects that draw from and add to the work of many other mathematicians, and they're open-ended and ongoing projects rather than self-contained, contrived homework problems.

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It will help them increase their reputation, publish papers, get a Ph. D. (if they don't already have one), and get a better job.

Just don't count on this happening to you.

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    $\begingroup$ +1, only because I can't underline and set in bold the last line. (Well, I suppose I could, but I'm not going to.) $\endgroup$ – anomaly Aug 27 '16 at 23:56
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One of the things many mathematicians do is solve old problems and invent new mathematics. It's a part of a university professor's job, along with his or her teaching. For people who choose that career, the excitement of discovery is its own reward. Then they write about it for others to read. Occasionally a very talented mathematician solves a famous problem; then there might be other benefits - a prize, a cash award, a better job.

Some mathematicians work in industry; their job is to solve problems that come up in the company's business.

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It depends what you mean by a "hard problem".

Perhaps you mean a problem that was in some math competition, and which only a few competitors were able to solve.

At the other extreme, maybe you mean a famous problem that has been unsolved for decades (or even centuries), like the Millennium Problems.

For solving the first kind of problem, your rewards will be small. It's unlikely to affect your job prospects in either industry or academia.

For solving the second kind of problem, your rewards would probably be quite significant in academia. You might well get a nice full professorship at a prestigious university. Rewards in industry would not be significant. To most industrial folks, solving a Millenium Problem is a general indication of talent and dedication, but they'd say that (in general) Millenium Problems are not relevant to their business, or that you are "over-qualified" for the positions they have available.

If you solve a Millenium Problem, you get a million dollars. With $1M in your pocket, maybe you don't need to worry about finding a good job any more.

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