**Primary question**: What great mathematics are we missing out on because of language barriers?
Please post interesting results, pursuits, and branches in mathematics that have not been translated to English. Make sure to mention the language(s) they are accessible in!
**note**: this question (and the further two) are about specific theorems or branches in mathematics, not about bringing up 'general answers'. Please post the (original language) names of specific theorems, or tell about theories/branches in mathematics worked on by specific groups with the name of the group and a simple note how to contact the group (a name + university name, or www address, or email address suffice).
And as a **secondary question**:
Please post examples of where mathematical thought is more easily expressed or more intuitive in a language other than English, rather than in English.
As a **ternary question**:
Please post examples of mathematical theories in history that have stalled due to unnatural or awkward spoken-language associated with the theory, while counterparts in another language have flourished.
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I know this question is English-centric, but if I were posting this on a German- (Russian-, Spanish-, Japanese-) speaking forum I would make it German-centric; on the other hand English is now the standard into which people translate their papers.
*One historical example of a big chunk of maths that never made it to English is Grothendieck's EGA.*
A co-question to this question has been asked here:
[What are good non-English languages for mathematicians to know?][1]
However, I disagree with the comments that languages other than English are only good for dead maths; pending research in every country happens in the local language, and that is a huge amount of knowledge to wave away.
A similar question but not directly stated in this form is here:
[Books you would like to see translated into English][2]
This question is a chance for the native (as well as second-language) speakers of many beautiful languages to tell about mathematical ideas, concepts, theories and results that have not yet transcended the language they were first written in. It is also a place to reflect on expressing specific mathematical concepts in very elegant ways that you think surpass the way we are thinking of them in English. I don't know if this is against the rules, but if not, do not hesitate to post examples in the language you're talking about.
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**Motivation and background:**
The recent question on what books we would like to have translated to English has revived an idea, or question, that I had about the way we learn and propagate ideas in mathematics. What knowledge, and more specifically mathematics (just to be on topic) are we missing out on because we don't know the less-commonly-taught second languages?
One person might say that English is the main language currently in which we are publishing papers. For example [Mathematics as a language (Wikipedia)][3]
> It is interesting to note that there are very few cultural dependencies or barriers in modern mathematics.
This might be true, to some extent. We all have seen those *bad* papers - many of us have commited the crimes - of submitting papers with broken English, explaining delicate concepts with the subtlety of a jackhammer. The lingual density of mathematics is immense and, since the spoken word is much more precise, we enhance it by using small nuances that we exaggerate (e.g. contains/consists of) to fit more information into the language that we speak. Before this paper makes it to an English-speaking journal a lot of mathematics is lost: it might be lost in translation; maybe the author didn't have enough time to translate everything; perhaps the author brought up some interesting adages that didn't work in English; maybe their first language allowed a specific 'slang' that made the concepts much easier to talk about. Most importantly, before a paper is submitted, a big, big amount of work happens - you will not learn of it before the select results are published; for one thing, it is a lot of time; for another, we all know that sometimes the most interesting mathematics stay hidden because they somehow didn't make the cut. Finally, maybe the research group did not publish to English because their work was meant to support other research in their country; or they just didn't want to bother, being happy with reaching their local environment. This is a place to bring up this sort of research.
In this question we are talking about understanding on a level much higher than 'being able to apply the theorems and formulas'. We are also talking about the 'enlightenment'. A lot of - maybe most? - mathematical thought is encoded in the every-day language being used to describe it, which can be more or less elegant. The fitness of this language to the purpose of the concept can make a big difference - compare the Newtonian school of Calculus becoming stalled because they would not want to forgo the 'dot notation' ($\dot{x}$) that is now largely abandoned and limited only to papers in mechanics. Compare [Origins of Mathematical Symbols/Names][4]
A concept can be explained in raw, dry definitions and formulas using thousands of sentences, or it can be explained in a swift, elegant way because the language has got just the right logical constructs and subtle interactions between linguistic concepts to express the logical constructs and interactions between the concepts in mathematical theory we are learning. Compare [Examples of great mathematical writing][5]
Related but orthogonal questions that might explain the nature of the problem at hand; further reading:
[MO: What are some good resources for mathematical translation?][6]
[MO: What’s so great about blackboards?][7]
[MO: japanese/chinese for mathematicians?][8]
[Towards a New Model of Bilingual Mathematics Teaching: the case of China][9]
[Mathematics is Not a Universal Language - Tara N. Tevebaugh; Teaching Children Mathematics, Vol. 5, December 1998][10] - children are having problems understanding simple, basic mathematics across language barriers; what can be said about immensely more complex ideas that we try to juggle? Concepts can be expressed in mathematical notation; the motivation of those concepts cannot be.
Thanks!
P.S. This is one of my first few posts on MO. I welcome any comments that can make my contributions better.
[1]: https://mathoverflow.net/questions/8056/what-are-good-non-english-languages-for-mathematicians-to-know
[2]: https://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/20861
[3]: http://en.wikipedia.org/wiki/Mathematics_as_a_language
[4]: https://mathoverflow.net/questions/8295/origins-of-mathematical-symbols-names
[5]: https://mathoverflow.net/questions/358/examples-of-great-mathematical-writing
[6]: https://mathoverflow.net/questions/9451/what-are-some-good-resources-for-mathematical-translation
[7]: https://mathoverflow.net/questions/5936/whats-so-great-about-blackboards-closed
[8]: https://mathoverflow.net/questions/3237/japanese-chinese-for-mathematicians
[9]: http://www.informaworld.com/smpp/content~content=a739495006&db=all
[10]: http://www.questia.com/PM.qst?a=o&d=5002303489