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Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.

Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces." She died July 15, 2017.

I'm not expert in these areas of mathematics, but I am eager to know her main ideas and the importance of her results.

Can one draw a general picture of her works?

(Any expository reference will be appreciated).

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    $\begingroup$ math.harvard.edu/~ctm/papers/home/text/papers/icm14/icm14.pdf $\endgroup$
    – Vesselin Dimitrov
    Commented Mar 6, 2015 at 6:32
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    $\begingroup$ In general, if you want to know about a Fields Medalist's (pre-medal) work, the Laudationes is a good place to start, and is available at the IMU website. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 6, 2015 at 6:46
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    $\begingroup$ Wright's introductory notes are very good source in my opinion - arxiv.org/abs/1411.1827 , but in general, as her recent work (the measure classification for translations surfaces with Eskin) combines 2 complicated notions (translation surface, and homogeneous dynamics and assorted measure classification results such as the Benoist-Quint theorem) it's hard to get a real grasp of the matter. I know even some experts in dynamics who simply don't get the geometric part of her work about translation surfaces. $\endgroup$
    – Asaf
    Commented Mar 6, 2015 at 6:58
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    $\begingroup$ I also have a shorter introduction, which is intended to be easy reading and will appear in the Bulletin of the AMS. web.stanford.edu/~amwright/BilliardsToModuli.pdf. $\endgroup$
    – Alex
    Commented Jul 28, 2015 at 16:25

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A very good expository article (in Farsi) on recent work of Maryam Mirzakhani can be found here. (PDF)

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    $\begingroup$ Indeed, that looks like a nice article with beautiful pictures. Now I just need to find a couple of hours to learn Farsi ... $\endgroup$
    – Neil Strickland
    Commented Mar 6, 2015 at 8:42
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    $\begingroup$ Dear Mostafa, Thank you so much for your answer. $\endgroup$
    – user68866
    Commented Mar 6, 2015 at 9:48
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    $\begingroup$ This note is really nice specially with interesting pictures. Fortunately I know Farsi a little. $\endgroup$
    – user68866
    Commented Mar 6, 2015 at 9:59
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    $\begingroup$ @Neil Strickland: it's only mathematical Farsi, so you'll pick it up in no time. $\endgroup$
    – R.P.
    Commented Mar 6, 2015 at 13:31
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    $\begingroup$ @NeilStrickland: technical comment: it seems that Eaman Eftekhary's careful typesetting, combined with the largest engine's services and current operating system infrastructure, makes it possible to more or less machine-translate the article, piecemeal at least, using ordinary copy and paste. For example, on the system I am currently on, Eftekhary's article allows me to just copy, from page ١٠ , the passage قضیۀ میرزاخانr در مورد رفتار مجانبی, which the largest engine translates as "Mirza Khan's case on asymptomatic behavior", the latter being sort-of understandable.Your mileage may vary. $\endgroup$
    – Peter Heinig
    Commented Sep 20, 2017 at 11:19
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Alex Wright has written an introductory survey on translation surfaces and orbit closures, which may be helpful in appreciating why Mirzakhani's most recent results on orbit closures of translation surfaces are so important.

Alex Wright - Translation surfaces and their orbit closures: An introduction for a broad audience

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A nice introduction, accesible to the layman (at least to a layperson with science studies), is published by the e-journal Quanta Magazine. See

A Tenacious Explorer of Abstract Surfaces by Erica Klarreich.

Also at Wired.

Update: There is a detailed description of her work in this paper by Curtis McMullen.

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You might be interested in
Anton Zorich's paper "The Magic Wand Theorem of A. Eskin and M. Mirzakhani"
which is the English version of
Anton Zorich's paper "Le théorème de la baguette magique de A. Eskin et M. Mirzakhani"
published in French in volume 142 of La Gazette des Mathématiciens.
This volume also has papers about other recipients of the 2014 Fields Medal.

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This is an article in German on Mirzakhani's early (published at the age of 19) paper on graph coloring, where she gave an example of a 3-colorable, but not 4-list-colorable graph: https://www.mathematik.de/images/Blog/Dokumente/dmvm-2015-0015.pdf

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Following up on Tom Church's answer, Alex Wright has recently written an overview for Mirzakhani's work on surfaces.

A tour through Mirzakhani's work on moduli spaces of Riemann surfaces

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