# Maryam Mirzakhani

Maryam Mirzakhani (Persian: مریم میرزاخانی‎, pronounced [mæɾˈjæm miːɾzɑːxɑːˈniː]; 12 May 1977 – 14 July 2017) was an Iranian[5][6][7][1] mathematician and a professor of mathematics at Stanford University.[8][9][10] Her research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.[1]

Maryam Mirzakhani
Mirzakhani at the 2014 International Congress of Mathematicians in Seoul, where she received her Fields Medal
Native name Persian: مریم میرزاخانی
Born 12 May 1977
Died 14 July 2017 (aged 40)
Stanford, California, U.S.
Nationality Iranian[1][2]
Education Sharif University of Technology (BSc)
Harvard University (PhD)
Spouse(s) Jan Vondrák
Children 1
Awards
Scientific career
Fields Mathematics
Institutions
Thesis Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves (2004)

On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics.[11][12] Thus, she became both the first woman and the first Iranian to be honored with the award.[13] The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".[14]

On 14 July 2017, Mirzakhani died of breast cancer at the age of 40.[15]

## Early life and educationEdit

Mirzakhani was born on 12 May 1977[5] in Tehran, Iran. Her father Ahmad is an electrical engineer.[7] She attended Tehran Farzanegan School there, part of the National Organization for Development of Exceptional Talents (NODET). In 1994, Mirzakhani achieved the gold medal level in the International Mathematical Olympiad, the first female Iranian student to do so.[16] In the 1995 International Mathematical Olympiad, she became the first Iranian student to achieve a perfect score and to win two gold medals.[17][18][19]

She obtained her BSc in mathematics in 1999 from the Sharif University of Technology. She then went to the United States for graduate work, earning her Ph.D. in 2004 from Harvard University, where she worked under the supervision of the Fields Medalist Curtis T. McMullen.[20] At Harvard she is said to have been "distinguished by ... determination and relentless questioning", despite not being a native English-speaker. She used to take her class notes in Persian.[21]

## CareerEdit

Mirzakhani was a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University.[22] In 2009, she became a professor at Stanford University.[14][23]

## Research workEdit

Maryam Mirzakhani, August 2014

Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. Mirzakhani’s early work solved the problem of counting simple closed geodesics on hyperbolic Riemann surfaces by finding a relationship to volume calculations on moduli space. Geodesics are the natural generalization of the idea of a "straight line" to "curved spaces". Slightly more formally, a curve is a geodesic if no slight deformation can make it shorter. Closed geodesics are geodesics which are also closed curves—that is, they are curves that close up into loops. A closed geodesic is simple if it does not cross itself.

A previous result, known as the “prime number theorem for geodesics”, established that the number of closed geodesics of length less than L grows exponentially with L—it is asymptotic to ${\displaystyle e^{L}/L}$ . However, the analogous counting problem for simple closed geodesics remained open, despite being “the key object to unlocking the structure and geometry of the whole surface,” according to University of Chicago topologist Benson Farb.[24] Mirzakhani’s 2004 PhD thesis solved this problem, showing that the number of simple closed geodesics of length less than L is polynomial in L. Explicitly, it is asymptotic to ${\displaystyle cL^{6g-6}}$ , where g is the genus (roughly, the number of “holes”) and c is a constant depending on the hyperbolic structure. This result can be seen as a generalization of the theorem of the three geodesics for spherical surfaces.[25][26]

Mirzakhani solved this counting problem by relating it to the problem of computing volumes in moduli space—a space whose points correspond to different complex structures on a surface genus g. In her thesis, Mirzakhani found a volume formula for the moduli space of bordered Riemann surfaces of genus g with n geodesic boundary components. From this formula followed the counting for simple closed geodesics mentioned above, as well as a number of other results. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space.[8][27]

Her subsequent work focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.[28] One can construct a simple earthquake map by cutting a surface along a finite number of disjoint simple closed geodesics, sliding the edges of each of these cut past each other by some amount, and closing the surface back up. One can imagine the surface being cut by strike-slip faults. An earthquake is a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them.

In 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal.[29][30] The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990s.[30] The International Mathematical Union said in its press release that "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."[30]

### Awarding of Fields MedalEdit

Four Fields medallists left to right Artur Avila, Martin Hairer (at back), Maryam Mirzakhani (with her daughter Anahita) and Manjul Bhargava at the ICM 2014 in Seoul

Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".[31] The award was made in Seoul at the International Congress of Mathematicians on 13 August.[32] At the time of the award, Jordan Ellenberg explained her research to a popular audience:

[Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables ... This isn't the kind of thing you do to win at pool, but it's the kind of thing you do to win a Fields Medal. And it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there.[33]

In 2014, President Hassan Rouhani of Iran congratulated her for winning the topmost world mathematics prize.[34]

Mirzakhani has an Erdős number of 3.[35]

## Personal lifeEdit

In 2008, Mirzakhani married Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who currently is an associate professor at Stanford University.[36][37] They have a daughter named Anahita.[38] Mirzakhani lived in Palo Alto, California.[39]

Mirzakhani described herself as a "slow" mathematician, saying that "you have to spend some energy and effort to see the beauty of math." To solve problems, Mirzakhani would draw doodles on sheets of paper and write mathematical formulas around the drawings. Her daughter described her mother's work as "painting".[40][41]

She declared:

I don't have any particular recipe [for developing new proofs] ... It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck, you might find a way out.[40]

## Death and legacyEdit

Mirzakhani was diagnosed with breast cancer in 2013.[42] In 2016, the cancer spread to her bones and liver,[40][43] and she died on 14 July 2017 at the age of 40 at Stanford Hospital in Stanford, California.[40][44][7][45]

Iranian president Hassan Rouhani and other officials published condolence messages and praised Mirzakhani's scientific achievements. Rouhani said in his message that "the unprecedented brilliance of this creative scientist and modest human being, who made Iran's name resonate in the world's scientific forums, was a turning point in showing the great will of Iranian women and young people on the path towards reaching the peaks of glory and in various international arenas."[46] Sharif University of Technology, the place where Mirzakhani studied, announced that its faculty of mathematics will be renamed to "Mirzakhani".[citation needed]

Upon her death, several Iranian newspapers, along with Iranian President Hassan Rouhani, broke taboo and published photographs of Mirzakhani with her hair uncovered, a gesture that was widely noted in the press and on social media.[13][47][48][49] Mirzakhani's death has also renewed debates within Iran regarding matrilineal citizenship for children of mixed-nationality parentage; Fars News Agency reported that, on the heels of Mirzakhani's death, 60 Iranian MPs urged the speeding up of an amendment to a law that would allow children of Iranian mothers married to foreigners to be given Iranian nationality, in order to make it easier for Mirzakhani's daughter to visit Iran.[13][48][50][51][52][53]

Numerous obituaries and tributes were published in the days following Maryam Mirzakhani's death.[54][55][56][57][58]

On February 2, 2018, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Maryam Mirzakhani.[59]

## ReferencesEdit

1. Mirzakhani, Maryam. "Curriculum Vitae" (PDF). Archived from the original (PDF) on 24 November 2005. Retrieved 13 August 2014.
2. ^ Valette, Alain. "The Fields Medalists 2014" (PDF). Neuchâtel, Switzerland: Institut de mathématiques, Université de Neuchâtel. Retrieved 15 July 2017.
3. ^ Jonathan, Webb (2014). "First female winner for Fields maths medal". BBC News. BBC News. Retrieved 13 August 2014.
4. ^ "Private Funeral of Professor Mirzakhani to be held in the United States", Iranian Students News Agency (in Persian), 19 July 2017, 96042715699, retrieved 19 July 2017
5. ^ a b "وبسایت رسمی مریم میرزاخانی". mmirzakhani.com (in Persian). Retrieved 2018-09-06.
6. ^ "Maryam Mirzakhani, first woman to win maths' Fields Medal, dies". BBC News. bbc.com. 2014-08-12. Retrieved 2017-07-16.
7. ^ a b c Chang, Kenneth (16 July 2017). "Maryam Mirzakhani, only woman to win a Fields Medal, dies at 40". The New York Times.
8. ^ a b Mirzakhani, Maryam (2007). "Weil-Petersson volumes and intersection theory on the moduli space of curves" (PDF). Journal of the American Mathematical Society. 20 (1): 1–23. Bibcode:2007JAMS...20....1M. doi:10.1090/S0894-0347-06-00526-1. MR 2257394.
9. ^ Mirzakhani, Maryam (January 2007). "Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces". Inventiones Mathematicae. 167 (1): 179–222. Bibcode:2006InMat.167..179M. doi:10.1007/s00222-006-0013-2. ISSN 1432-1297.
10. ^ "Report of the President to the Board of Trustees". Stanford University. 9 April 2008. Retrieved 12 August 2014.
11. ^ "President Rouhani Congratulates Iranian Woman for Winning Math Nobel Prize". Fars News Agency. 14 August 2014. Retrieved 14 August 2014.
12. ^ a b "IMU Prizes 2014". International Mathematical Union. Retrieved 12 August 2014.
13. ^ a b c Iran correspondent Saeed Kamali Dehghan (2017-07-16). "Maryam Mirzakhani: Iranian newspapers break hijab taboo in tributes". The Guardian. ISSN 0261-3077. Retrieved 2017-07-18.
14. ^ a b Sample, Ian (13 August 2014). "Fields Medal mathematics prize won by woman for first time in its history". The Guardian. Retrieved 9 June 2016.
15. ^ "Maryam Mirzakhani's Pioneering Mathematical Legacy". The New Yorker. 2017-07-17. Retrieved 2017-07-18.
16. ^ "The 57th International Mathematical Olympiad Successfully Completed in Hong Kong". Archived from the original on 24 July 2016. Retrieved 20 July 2017.
17. ^
18. ^ "Iranian woman wins maths' top prize". New Scientist. 12 August 2014. Retrieved 13 August 2014.
19. ^ Newhall, Marissa (13 September 2005). "'Brilliant' minds honored: Maryam Mirzakhani". USA Today.
20. ^ "Maryam Mirzakhani". The Mathematics Genealogy Project. Genealogy.math.ndsu.nodak.edu. Retrieved 21 October 2017.
21. ^ "Maryam Mirzakhani, Stanford mathematician and Fields Medal winner, dies". news.stanford.edu. 2017-07-15. Retrieved 21 October 2017.
22. ^ Maryam Mirzakhani's publications indexed by the Scopus bibliographic database. (subscription required)
23. ^ Juris, Yvonne (16 July 2017). "Maryam Mirzakhani, first woman to receive the prestigious Fields Medal, dies at the age of 40 after breast cancer battle". People Magazine.
24. ^ "Maryam Mirzakhani Is First Woman Fields Medalist | Quanta Magazine". Quanta Magazine. Retrieved 2018-09-25.
25. ^ Mirzakhani, Maryam (2008). "Growth of the number of simple closed geodesics on hyperbolic surfaces". Annals of Mathematics. 168 (1): 97–125. doi:10.4007/annals.2008.168.97. MR 2415399. Zbl 1177.37036.
26. ^ Kehoe, Elaine (April 2013). "Notices of the AMS" (PDF). Notices of the AMS. Retrieved September 25, 2018.
27. ^ "2014 Fields Medals" (PDF). Notices of the AMS. 61 (9): 1079–1081. October 2014.
28. ^ Mirzakhani, M. (2008). "Ergodic Theory of the Earthquake Flow". International Mathematics Research Notices. 2008. doi:10.1093/imrn/rnm116. MR 2416997.
29. ^ Eskin, Alex; Mirzakhani, Maryam; Mohammadi, Amir (2015). "Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space". Annals of Mathematics. 182 (2): 673–721. arXiv:1305.3015. doi:10.4007/annals.2015.182.2.7.
30. ^ a b c "The Work of Maryam Mirzakhani" (PDF) (Press release). International Mathematics Union. Retrieved 15 August 2014.
31. ^ "IMU Prizes 2014 citations". International Mathematical Union. Retrieved 12 August 2014.
32. ^ Carey, Bjorn (12 August 2014). "Stanford's Maryam Mirzakhani wins Fields Medal". Stanford News. Retrieved 13 August 2014.
33. ^ Ellenberg, Jordan (13 August 2014). "Math Is Getting Dynamic". Slate.
34. ^ "President hails Prof Mirzakhani, winner of topmost world math prize". Official Site of the President of The Islamic Republic of Iran. 13 August 2014. Retrieved 19 August 2014.
35. ^ "Collaboration paths to Paul Erdős". The Erdős Number Project.
36. ^ "بیوگرافی مریم میرزاخانی؛ ستاره پرفروغ دنیای ریاضیات [Biography Maryam Mirzakhani; the best-selling star of the world of mathematics]" (in Persian). Zoomit. 24 July 2011.
37. ^ "Jan Vondrák, CV" (PDF). Stanford University. Retrieved 14 July 2017.
38. ^ "A Tenacious Explorer of Abstract Surfaces", simonsfoundation.org. Retrieved 13 April 2014.
39. ^ Putic, George (13 August 2014). "Iranian-American Woman Wins Top Mathematics Prize". Voice of America. Retrieved 18 July 2017.
40. ^ a b c d Myers, Andrew; Carey, Bjorn (15 July 2017). "Maryam Mirzakhani, Stanford mathematician and Fields Medal winner, dies". Stanford News. Retrieved 17 July 2017.
41. ^ Jacobson, Howard (29 July 2017). "The world has lost a great artist in mathematician Maryam Mirzakhani". The Guardian. Retrieved 21 October 2017.
42. ^ "PressTV-Iranian math genius battles cancer at US hospital". Presstv.ir. Retrieved 2017-07-16.
43. ^ France-Presse, Agence (2017-07-15). "Sorrow as Maryam Mirzakhani, the first woman to win mathematics' Fields Medal, dies aged 40". The Telegraph. Retrieved 15 July 2017.
44. ^ "Maryam Mirzakhani died" (in Persian). Mehr news Agancy. 15 July 2017.
45. ^ "مریم میرزاخانی، ریاضیدان برجسته ایرانی درگذشت [Maryam Mirzakhani, a prominent Iranian mathematician, dies]". BBC News فارسی (in Persian). BBC Persian. 2017-07-15. Retrieved 15 July 2017.
46. ^ "Iranian math genius Mirzakhani passes away". 15 July 2017.
47. ^ Samuel, Sigal. "Why Iran Broke Its Strict Hijab Rules for the 'Queen of Math'". The Atlantic. Retrieved 2017-07-18.
48. ^ a b "Iranian Media Break Hijab Taboo in Tributes to Maryam Mirzakhani". thewire.in. Retrieved 2017-07-19.
49. ^ "Iranian Press Flouts Hijab Rules In Death Tributes To Trailblazing Maths Genius". HuffPost UK. 2017-07-17. Retrieved 2017-07-19.
50. ^ "اصلاح قانون تابعیت عمل به وصیتنامه مرحوم میرزاخانی نابغه ریاضی جهان را ..." Kodoom.com (in Persian). Retrieved 2017-07-19.
51. ^ Harris, Chris (2017-07-17). "How death of maths genius Mirzakhani is breaking taboos for women in Iran". euronews. Retrieved 2017-07-19.
52. ^ "Maryam Mirzakhani, First Woman To Win The Fields Medal, Dies At 40". IFLScience. Retrieved 2017-07-19.
53. ^ "Iranian newspapers break hijab taboo in tribute to Maryam Mirzakhani". Siasat.com. 2017-07-17. Retrieved 2017-07-19.
54. ^ "A Tribute to Maryam Mirzakhani". American Mathematical Society. 18 July 2017. Retrieved 3 August 2017.
55. ^ Roberts, Siobhan (17 July 2017). "Maryam Mirzakhani's Pioneering Mathematical Legacy". The New Yorker. Retrieved 3 August 2017.
56. ^ Chas, Moira (24 July 2017). "The Beautiful Mathematical Explorations of Maryam Mirzakhani". Quanta Magazine. Retrieved 3 August 2017.
57. ^ Halpern, Paul (1 August 2017). "Maryam Mirzakhani, A Candle Illuminating the Dark". Forbes. Retrieved 3 August 2017.
58. ^ Wright, Alex (2017-08-25). "Maryam Mirzakhani (1977–2017)". Science. 357 (6353): 758. doi:10.1126/science.aao6074. ISSN 0036-8075. PMID 28839063.
59. ^ "China lofts earthquake research craft with cluster of smaller satellites – Spaceflight Now". spaceflightnow.com. Retrieved 2018-02-04.
60. ^ "Interview with Research Fellow Maryam Mirzakhani" (PDF). Oxford University. 2008.
61. ^ a b American Mathematical Society. Retrieved 6 January 2009
62. ^ "ICM Plenary and Invited Speakers since 1897". International Congress of Mathematicians. Retrieved 13 August 2014.
63. ^ "2013 Simons Investigators Awardees | Simons Foundation". Simonsfoundation.org. Retrieved 2017-08-03.
64. ^ Gibney, E.; Leford, H.; Lok, C.; Hayden, E.C.; Cowen, R.; Klarreich, E.; Reardon, S.; Padma, T.V.; Cyranoski, D.; Callaway, E. (18 December 2014). "Nature's 10 Ten people who mattered this year". Nature. 516 (7531): 311–319. Bibcode:2014Natur.516..311.. doi:10.1038/516311a. PMID 25519114.
65. ^ "2014 Clay Research Awards – Clay Mathematics Institute". Claymath.org. Retrieved 21 October 2017.
66. ^ Larousserie, David (12 August 2014). "Médaille Fields de mathématiques : une femme promue pour la première fois". Le Monde (in French). Retrieved 13 August 2014.
67. ^ "Quinze nouveaux associés étrangers à l'Académie des sciences" (PDF). Institute de France Académie des Sciences. Retrieved 16 July 2017.
68. ^ Newly Elected, American Philosophical Society, April 2015, archived from the original on 16 August 2015, retrieved 28 August 2015
69. ^ "National Academy of Sciences Members and Foreign Associates Elected". Retrieved 2016-05-05.
70. ^ Maryam Mirzakhani elected to American Academy of Arts and Sciences, Department of Mathematics, Stanford University, May 2017, retrieved 2017-05-06
71. ^ "321357 Mirzakhani (2009 MM)". Minor Planet Center. Retrieved 6 February 2018.
72. ^ "MPC/MPO/MPS Archive". Minor Planet Center. Retrieved 6 February 2018.