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I see that many PhD programs require students to demonstrate students’ abilities to translate English mathematics papers to two of French, Russian, and German and different papers in those languages in two of those languages to back to English. I eventually plan on doing work in algebraic geometry and that end of mathematics, so I know that being able to read and write French would be useful to some extent.

I do have significant exposure to two other Romance languages, but I am certainly not at the native level in them—I can read both quite well but can’t really write well in the other (speaking seems irrelevant here)—and it appears that I would have to learn another language in addition to French. I know that I have a starting point with French, but German and Russian are a little different from any languages I know even if English is a Germanic language, so I am a little scared.

How many semesters of study should I put into language study? How do I balance this with taking advanced courses in the area of mathematics I want to do research in as well as undergraduate research itself? Up to how advanced a level do most people aiming for top 10 math PhD programs usually take? And how bad is it that I did not study French, German, or Russian in high (secondary) school? Or is it that people in these programs are usually motivated and intelligent enough to learn to read and write in languages on the side without guidance or too much interfere with math study?

I saw similar posts on the academia.stackexchange site, but none of them seemed to answer my questions.

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I have never ever ever ever seen/heard of somebody's Ph.D being in jeopardy just because they couldn't fulfill the foreign language. In my opinion this requirement is archaic and unnecessary. In fact, just a few weeks ago, my dept abandoned this requirement. I wouldn't worry too much about this. –  Fixed Point Dec 12 '13 at 10:37
    
It can actually be helpful studying a paper in a language you do not really understand, because that prevents you from skim-reading and missing important points. I have managed to follow a book chapter in Russian, having never learned the language at all, by transliterating every letter from Cyrillic and with a dictionary to hand. Going the other way would be harder. –  HTFB Dec 12 '13 at 10:48
    
One should note that the ability to read and understand math papers in those languages (such as knowiung that field is Körper and not Feld in German) is very different from any proficiency in the language itself (such as enjoying Goethe's poems). Interestingly, there are specific courses for that purpose - I have a book "Russian for the Mathematician" at home which is already funny by the fact that it uses verbs like интегрировать (to integrate) and Число (the number) as conjugation and declination examples. –  Hagen von Eitzen Dec 12 '13 at 10:53
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