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What is the possibility for a computer scientist to change his field into a pure mathematician ? and what's the most smooth way to do it ? any examples for people who could make it ?

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    $\begingroup$ This question would be better suited on the hopefully-soon-to-be-launched Academia SE: area51.stackexchange.com/proposals/16617/academia $\endgroup$ – Anthony Labarre Feb 11 '12 at 8:14
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    $\begingroup$ @TsuyoshiIto: presumably the most smooth way is when the transition is in $C^\infty$. $\endgroup$ – Dave Clarke Feb 11 '12 at 16:33
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    $\begingroup$ infidel! cs=>math ask on mathoverflow. we will address queries here on the opposite direction :p $\endgroup$ – vzn Feb 11 '12 at 19:42
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    $\begingroup$ Theoretical computer scientists are already mathematicians. So you don't need to do anything! $\endgroup$ – Jeffε Feb 12 '12 at 2:32
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    $\begingroup$ @JɛffE: If I remember correctly, a mathematician is said to be pure if all its facets have the same dimension. $\endgroup$ – Tsuyoshi Ito Feb 13 '12 at 15:02
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Theoretical computer scientists are already mathematicians. So you don't need to do anything!

Your later comment suggests that you're really asking about changing from one area of mathematics (like complexity theory) to another (like functional analysis). The only way to do it is to just do it. You want to be an analyst? Great! Just start doing analysis! Read analysis books and papers, solve analysis problems, talk to lots of analyst faculty and students, attend analysis seminars, go to analysis conferences, ask analysis questions on MathOverflow, and so on. Act like an analyst long enough (and well enough) and you'll become one.

In that respect, becoming a functional analyst is not so different from becoming anything else. Want to be a programmer? Program! Want to be a writer? Write! Want to be a painter? Paint! Want to be a functional analyst? Analyze functions! (Meanwhile, don't forget to earn enough money to eat, and do not blow off the stupid administrative hurdles.)

And yes, lots of people successfully change fields/careers. For example, Joan Birman started her academic career in physics, worked for several years as a systems analyst in the aircraft industry, spent several more years at home raising kids, went back to grad school in mathematics in her 40s, and became one of the most influential researchers in modern low-dimensional topology and geometric group theory. For other examples, see these questions on MathOverflow.

One last suggestion: You might find the transition somewhat smoother (possibly even real-analytic) if you drop your prejudices about "pure" versus "applied" mathematics. Forget about the labels; just do the work.

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    $\begingroup$ +1 for the first line of your answer. Not that the rest is not correct, but I think that's the answer here :-) $\endgroup$ – Janoma Feb 13 '12 at 13:54
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    $\begingroup$ Yes. "Just do it" and "Just do the work". But they always don't correlate well with "don't forget to earn enough money to eat, and do not blow off the stupid administrative hurdles". Sometimes you do a certain field for satisfaction and have to go to something else to eat. So assess your talents and make sure you will be ok to do something else to eat in case what you choose to learn is not giving sufficient (it is the case in theory cs and other phds in average ranked schools as well). Have a open mind to challenges particularly in math and have a lot of patience in learning advanced math. $\endgroup$ – v s Feb 13 '12 at 15:13

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