玛丽亚姆·米尔扎哈尼Maryam Mirzakhani

玛丽亚姆·米尔扎哈尼的父母是Ahmad Mirzakhani，一位电气工程师，和Zahra Haghighi。她出生于1977年5月12日，尽管有些参考资料错误地将5月3日作为她的生日。米尔扎哈尼这样谈到她的家庭（见[6]或[7]）：-

我在一个有兄弟姐妹三人的家庭中长大。我的父母一直非常支持和鼓励我们。对他们来说，重要的是我们拥有有意义且令人满意的职业，但他们并不那么在意成功和成就。从很多方面来说，这对我来说是一个很好的环境，尽管当时正值两伊战争的艰难时期。我的哥哥是让我对科学产生兴趣的人。他常常告诉我他在学校学到的东西。

她在德黑兰长大的早年生活很艰难，因为两伊战争从1980年持续到1988年。米尔扎哈尼有着丰富的想象力，当她八岁时，她会编造关于一个女孩取得伟大成就的故事，比如成为市长或环游世界。大约在战争结束时，米尔扎哈尼完成了小学学业，并参加了德黑兰Farzanegan女子中学的考试。这所学校由伊朗国家杰出人才发展组织管理，旨在教育最聪明的学生。她说（见[6]或[7]）：-

我在进入中学后的第一周遇到了我的朋友Roya Beheshti。有一个与你志趣相投、帮助你保持动力的朋友是无价的。我们的学校靠近德黑兰一条满是书店的街道。我记得沿着这条拥挤的街道行走，走进那些书店，对我们来说是多么令人兴奋。我们不能像人们在这里的书店里通常那样浏览书籍，所以我们最终会买很多随机的书。

Roya Beheshti 和米尔扎哈尼一样，出生于1977年。她现在是美国圣路易斯华盛顿大学的数学副教授，研究兴趣在代数几何。

由于书很便宜，随机购买并不是一个糟糕的策略。米尔扎哈尼此时想，她愿意成为一名作家并写这样的书。她最喜欢的电视节目是传记，像玛丽·居里和海伦·凯勒这样的人的故事激励了她。在法尔扎内甘中学的第一年，她的数学成绩并不特别好，她的老师告诉她，她在这门学科上并不特别有天赋。这对她的自信心是一个打击，她对数学失去了兴趣。然而，在第二年，她遇到了一位不同的数学老师，这位老师鼓励她，这使她展现出巨大的天赋。

米尔扎哈尼和她的朋友Roya Beheshti都从Farzanegan中学升入Farzanegan高中。她们都找到了一份六道数学奥林匹克问题的副本，米尔扎哈尼设法做出了其中三道。受此鼓励，她和朋友一起去找校长，问她能否为她们安排数学解题课。这些课在最有天赋的男生学校里进行，但女生没有。Farzanegan高中的校长非常鼓励，尽管从来没有女生参加过伊朗数学奥林匹克队，还是安排了课程。米尔扎哈尼说[9]：-

校长性格非常坚强。如果我们真的想要什么，她就会让它实现。她的心态非常积极乐观——“你能做到，即使你会是第一个。”我认为这对我的人生产生了很大影响。

米尔扎哈尼和她的朋友Roya Beheshti在1994年入选伊朗数学奥林匹克队。那年的国际比赛在香港举行，米尔扎哈尼得了42分中的41分，获得金牌。Beheshti获得银牌。1995年，米尔扎哈尼再次成为伊朗数学奥林匹克队成员。这次国际比赛在加拿大多伦多举行，米尔扎哈尼得了42分中的42分，再次获得金牌。

1995年，米尔扎哈尼在谢里夫理工大学开始学习数学，由IPM奖学金资助。这所位于德黑兰的大学建于1966年，是伊朗物理科学领域的领先大学。她说（见[6]或[7]）：-

我在谢里夫大学遇到了许多鼓舞人心的数学家和朋友。我在数学上花的时间越多，就越兴奋。在谢里夫大学，我们和同学一起进行解题课和非正式读书小组。我在那里遇到的所有人的友谊和支持……在许多不同方面帮助了我很多。

米尔扎哈尼 在本科期间就发表了论文。她与 E S Mahmoodian 合作，于 1995 年在德黑兰举行的“组合学进展”会议的 Proceeding 上发表了 Decomposition of complete tripartite graphs into 5-cycles。她还在 1996 年发表了 A small non-4-choosable planar graph，在 1998 年发表了 A simple proof of a theorem of Schur。

1998年2月，谢里夫理工大学的顶尖数学学生参加了在伊朗城市阿瓦士举行的一场比赛。米尔扎哈尼是这些学生之一。一辆载着学生从阿瓦士返回德黑兰的巴士打滑，坠入峡谷。七名学生和两名巴士司机在事故中丧生，但米尔扎哈尼幸存了下来。她于1999年从谢里夫理工大学毕业，获得学士学位。

从谢里夫大学获得学位后，米尔扎哈尼 前往美国，在哈佛大学攻读研究生。在那里，她开始参加 柯蒂斯·麦克马伦 的讨论班。柯蒂斯·麦克马伦 于 1998 年被任命为哈佛大学教授，同年他在柏林国际数学家大会上被授予 菲尔兹奖。柯蒂斯·麦克马伦 成为她的博士导师。他说 [9]：-

她有一种大胆的想象力。她会在脑海中构想出必定正在发生的事情的想象图景，然后来到我的办公室描述它。最后，她会转向我说：“对吗？”她认为我会知道，这总让我感到非常荣幸。

她的工作涉及 [11]：-

……双曲曲面上的闭测地线。这些是长度无法通过形变缩短的闭曲线。一个50多年前证明的如今已成为经典的定理，给出了一种精确估计长度小于某个界 L 的闭测地线数量的方法。闭测地线的数量随 L 指数增长……米尔扎哈尼 考察了当只考虑简单的、即不自交的闭测地线时，“测地线的素数定理”会发生什么。在这种情况下，行为非常不同：长度至多为 L 的测地线数量的增长不再是 L 的指数增长，而是 $L^{6g-6}$ 的量级，其中 g 是亏格。

哈佛大学于2003年授予米尔扎哈尼一项优异奖学金。她于2004年因其130页的学位论文Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves. 而获得博士学位。由于这篇杰出的学位论文，她于2009年被授予Leonard M和Eleanor B奥托·布卢门塔尔纯数学研究促进奖。引文如下：-

[米尔扎哈尼因其极具创造性、高度原创的学位论文而受到表彰]。这项工作结合了双曲几何、自守形式的“经典方法”以及辛约化等多样工具，在三个不同重要问题上取得了结果。这些结果包括安德烈·韦伊-Petersson体积的模空间上波恩哈德·黎曼曲面的递推公式、以长度表示的紧致双曲曲面上简单闭测地线数量的渐近性的确定，以及爱德华·威滕猜想（最初由马克西姆·孔采维奇建立）的新证明，该证明建立了模空间上交数的KdV递推。

2004年，她获得了哈佛大学的初级研究员职位，但拒绝了这一邀请，因为更好的机会在等着她。那一年，她获得了克莱研究奖学金，并被任命为普林斯顿大学数学助理教授。她说（见[6]或[7]）：-

[克莱奖学金]对我来说是一个很好的机会；我大部分时间都在普林斯顿度过，这是一次很棒的经历。克莱奖学金给了我自由去思考更难的问题、自由旅行，并与其他数学家交流。我是一个思考缓慢的人，必须花很多时间才能理清思路并取得进展。所以我真的很感激我不必匆忙地写出我的工作。

确实，她的下一篇论文是在她的学位论文发表三年后出现的，但有许多非常深刻的论文。这些论文是：Weil-Petersson volumes and intersection theory on the moduli space of curves（2007）；Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces（2007）；Random hyperbolic surfaces and measured laminations（2007）；Growth of the number of simple closed geodesics on hyperbolic surfaces（2008）；Ergodic theory of the earthquake flow（2008）；以及（与埃隆·林登施特劳斯合作）Ergodic theory of the space of measured laminations（2008）。

米尔扎哈尼的克莱研究奖学金持续到2008年，当时她离开普林斯顿，被任命为斯坦福大学数学教授。她遇到了Jan Vondrák，一位理论计算机科学家和应用数学家，他于2006年至2009年在普林斯顿大学担任博士后教学研究员。她嫁给了Vondrák，一位捷克人，拥有布拉格查理大学计算机科学博士学位和麻省理工学院应用数学博士学位，他们有一个女儿Anahita，出生于2011年。Vondrák于2016年1月被任命为斯坦福大学副教授。

正是在2006年，米尔扎哈尼开始与芝加哥大学的Alex Eskin合作。他们攻克了该领域最困难的问题之一，即推广柯蒂斯·麦克马伦在2003年发表的一个结果[11]：-

米尔扎哈尼与Alex Eskin以及部分与Amir Mohammadi合作，在理解模空间上另一个与模空间中测地线行为相关的动力系统方面取得了重大突破。模空间中的非闭测地线非常不规则，甚至病态，很难理解它们的结构以及它们在轻微扰动时如何变化。然而，米尔扎哈尼等人已经证明，模空间中的复测地线及其闭包实际上是出人意料地规则，而不是不规则或分形的。事实证明，虽然复测地线是用分析和微分几何定义的超越对象，但它们的闭包是用多项式定义的代数对象，因此具有某些刚性性质。

Eskin 谈到了米尔扎哈尼，说[9]：-

她非常乐观，这种乐观很有感染力。和她一起工作时，你会觉得解决那些起初看似无望的问题的机会要大得多。

有时 Eskin 感到悲观，但米尔扎哈尼没有：-

有时会遇到挫折，但她从不惊慌。

成功之后，她说：-

如果我们早知道事情会这么复杂，我想我们可能就放弃了。我不知道；其实，我不知道，我不会轻易放弃。

Eskin 与 米尔扎哈尼 于 2011 年发表了 Counting closed geodesics in moduli space。

2014 年，米尔扎哈尼 成为首位获得 约翰·查尔斯·菲尔兹 奖章的女性。该奖章由 国际数学联盟 于 2014 年 8 月 13 日在韩国首尔举行的国际数学家大会上颁发给她。颁奖词称该奖项是——

……表彰她对 波恩哈德·黎曼 曲面及其模空间的动力学与几何做出的杰出贡献。

在我们已经提到的荣誉之外，米尔扎哈尼 在其短暂的一生中还赢得了许多荣誉。她于 2013 年 1 月 10 日在圣迭戈获得露丝·利特尔·萨特数学奖。她于 2015 年当选 巴黎科学院 院士，2015 年当选美国哲学会会士，2016 年当选 国家科学院 院士，2017 年当选 美国艺术与科学院 院士。她曾是 2010 年国际数学家大会拓扑学与动力系统及常微分方程分会的受邀报告人，当时她作了题为 On Weil-Petersson volumes and geometry of random hyperbolic surfaces 的演讲。2014 年在韩国首尔举行的国际数学家大会上，她再次作为全会报告人发言。

甚至在 2014 年获得 约翰·查尔斯·菲尔兹 奖章之前，米尔扎哈尼 就已被诊断出患有乳腺癌。她继续从事数学工作，不仅产生了极具重要性的成果，还在此过程中开发了该领域研究人员在继续推进时将使用的工具。癌症扩散到她的肝脏和骨骼，她于 2017 年 7 月在加利福尼亚一家医院去世，使数学界失去了一颗最耀眼的明星，她年仅 40 岁，正处于创造力的巅峰。斯坦福大学校长 Marc Tessier-Lavigne 在她去世后表示 [12]：——

米尔扎哈尼 走得太早了，但她的影响将长存，因为她激励了成千上万的女性投身数学和科学。米尔扎哈尼 是一位杰出的数学理论家，也是一位谦逊的人，她接受荣誉时只希望这能鼓励他人追随她的道路。她作为学者和榜样的贡献重大而持久，斯坦福和全世界都将深切怀念她。

斯坦福大学Barbara Kimball Browning数学讲席教授Ralph L Cohen表示[12]：-

米尔扎哈尼是一位出色的同事。她不仅是一位才华横溢、无所畏惧的研究者，还是一位伟大的教师和极好的博士导师。米尔扎哈尼体现了身为数学家或科学家的全部意义：试图解决一个此前未被解决的问题，或理解某种此前未被理解的事物。这由深沉的求知欲所驱动，而每一次点滴的成功都带来巨大的喜悦与满足。米尔扎哈尼拥有我们这个时代最伟大的才智之一，她是一个了不起的人。人们将无比怀念她。

Maryam Mirzakhani's parents are Ahmad Mirzakhani, an electrical engineer, and Zahra Haghighi. She was born on 12th May 1977, though some references incorrectly give 3rd May as her birthday. Maryam said of her family (see [6] or [7]):-

I grew up in a family with three siblings. My parents were always very supportive and encouraging. It was important for them that we have meaningful and satisfying professions, but they didn't care as much about success and achievement. In many ways, it was a great environment for me, though these were hard times during the Iran-Iraq war. My older brother was the person who got me interested in science in general. He used to tell me what he learned in school.

Her young days were difficult ones growing up in Tehran since the Iran-Iraq war was fought from 1980 to 1988. Maryam had a great imagination and, when she was eight years old, she would make up stories about a girl who achieved great things, such as becoming mayor or travelling the world. About the time the war ended, Maryam completed her studies at elementary school and sat an examination for the Farzanegan middle school for girls in Tehran. This school, administered by Iran's National Organization for Development of Exceptional Talents, aimed to educate the brightest pupils. She said (see [6] or [7]):-

I met my friend Roya Beheshti the first week after entering middle school. It is invaluable to have a friend who shares your interests, and helps you stay motivated. Our school was close to a street full of bookstores in Tehran. I remember how walking along this crowded street, and going to the bookstores, was so exciting for us. We couldn't skim through the books like people usually do here in a bookstore, so we would end up buying a lot of random books.

Roya Beheshti was, like Maryam, born in 1977. She is now Associate Professor of Mathematics at Washington University in St Louis, United States, with research interests in algebraic geometry.

Since books were cheap, buying at random was not such a bad strategy. Maryam thought at this time that she would like to become an author and write such books. Her favourite television programmes were biographies and the stories of people like Marie Curie and Helen Keller inspired her. In this first year at Farzanegan middle school she did not do particularly well at mathematics and her teacher told her that she was not particularly talented in that subject. This was a blow to her confidence and she lost interest in it. However, in her second year she had a different mathematics teacher who encouraged her and this led to her showing great talent.

Both Maryam and her friend Roya Beheshti progressed from Farzanegan middle school to Farzanegan high school. They both found a copy of six Mathematical Olympiad problems and Maryam managed to do three of them. Encouraged by this she went with her friend to the school principal and asked her if she could arrange for them to have mathematical problem-solving classes. These happened in schools for the most talented boys, but not for the girls. The principal of Farzanegan high school was very encouraging and, even though no girls had ever taken part in the Iranian Mathematical Olympiad team, classes were arranged. Maryam said [9]:-

The principal of the school was a very strong character. If we really wanted something, she would make it happen. Her mindset was very positive and upbeat - that "you can do it, even though you'll be the first one." I think that has influenced my life quite a lot.

Both Maryam Mirzakhani and her friend Roya Beheshti made the Iranian Mathematical Olympiad team in 1994. The international competition was held that year in Hong Kong and Mirzakhani scored 41 out of 42 and was awarded a gold medal. Beheshti was awarded a silver medal. Again in 1995 Mirzakhani was a member of the Iranian Mathematical Olympiad team. This time the international competition was held in Toronto, Canada, and Mirzakhani scored 42 out of 42 and was again awarded a gold medal.

In 1995 Mirzakhani began her study of mathematics at Sharif University of Technology funded with an IPM fellowship. This university in Tehran, established in 1966, is the leading university in Iran for physical science. She said (see [6] or [7]):-

I met many inspiring mathematicians and friends at Sharif University. The more I spent time on mathematics, the more excited I became. At Sharif University, we had problem-solving sessions and informal reading groups with my classmates. The friendship and support of all the people I met there ... helped me a lot in many different ways.

Mirzakhani published papers while an undergraduate. Jointly with E S Mahmoodian, she published Decomposition of complete tripartite graphs into 5-cycles in the Proceeding of the conference 'Combinatorics advances' held in Tehran in 1995. She also published A small non-4-choosable planar graph in 1996 and A simple proof of a theorem of Schur in 1998.

In February 1998 the top mathematical students from Sharif University of Technology took part in a competition in the Iranian city of Ahwaz. Mirzakhani was one of these students. A bus bringing the students back from Ahwaz to Tehran skidded and crashed into a ravine. Seven students and two bus drivers died in the crash but Mirzakhani survived. She graduated from Sharif University of Technology in 1999 with a B.S.

After obtaining her degree from Sharif University, Mirzakhani went to the United States where she attended graduate school at Harvard University. There she started to attend Curtis McMullen's seminar. McMullen had been appointed to a professorship at Harvard University in 1998, the year in which he had been awarded a Fields Medal at the International Congress of Mathematicians in Berlin. McMullen became her doctoral advisor. He said [9]:-

She had a sort of daring imagination. She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, "Is it right?" I was always very flattered that she thought I would know.

Her work involved [11]:-

... closed geodesics on a hyperbolic surface. These are closed curves whose length cannot be shortened by deforming them. A now-classic theorem proved more than 50 years ago gives a precise way of estimating the number of closed geodesics whose length is less than some bound L. The number of closed geodesics grows exponentially with L ... Mirzakhani looked at what happens to the "prime number theorem for geodesics" when one considers only the closed geodesics that are simple, meaning that they do not intersect themselves. The behavior is very different in this case: the growth of the number of geodesics of length at most L is no longer exponential in L but is of the order of $L^{6g-6}$, where g is the genus.

Harvard University awarded Mirzakhani a Merit fellowship in 2003. She was awarded her doctorate in 2004 for her 130-page thesis Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves. For this outstanding thesis she was awarded the Leonard M and Eleanor B Blumenthal Award for the Advancement of Research in Pure Mathematics in 2009. The citation reads:-

[Mirzakhani is honoured] for her exceptionally creative, highly original thesis. This work combines tools as diverse as hyperbolic geometry, 'classical methods' of automorphic forms, and symplectic reduction to obtain results on three different important questions. These results include a recursive formula for Weil-Petersson volumes of moduli spaces of Riemann surfaces, a determination of the asymptotics of the number of simple closed geodesics on a hyperbolic surface in terms of length, and a new proof of Witten's Conjecture (originally established by Kontsevich) establishing the KdV recursion for the intersection numbers on moduli space.

In 2004 she was offered a junior fellowship at Harvard, but turned down the offer since something better awaited her. In that year she was awarded a Clay Research Fellowship and was appointed as an Assistant Professor of Mathematics at Princeton University. She said (see [6] or [7]):-

[The Clay fellowship] was a great opportunity for me; I spent most of my time at Princeton which was a great experience. The Clay Fellowship gave me the freedom to think about harder problems, travel freely, and talk to other mathematicians. I am a slow thinker, and have to spend a lot of time before I can clean up my ideas and make progress. So I really appreciate that I didn't have to write up my work in a rush.

Indeed her next papers appeared three years after her thesis was published, but there were many remarkably deep papers. These papers were: Weil-Petersson volumes and intersection theory on the moduli space of curves (2007); Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces (2007); Random hyperbolic surfaces and measured laminations (2007); Growth of the number of simple closed geodesics on hyperbolic surfaces (2008); Ergodic theory of the earthquake flow (2008); and (with Elon Lindenstrauss) Ergodic theory of the space of measured laminations (2008).

Mirzakhani's Clay Research Fellowship ran until 2008 when she left Princeton and was appointed as Professor of Mathematics at Stanford University. She had met Jan Vondrák, a theoretical computer scientist and applied mathematician who was a postdoctoral teaching fellow at Princeton University from 2006 to 2009. She married Vondrák, a Czech with a Ph.D. in Computer Science from the Charles University of Prague and a Ph.D. in Applied Mathematics from the Massachusetts Institute of Technology, and they have a daughter Anahita born in 2011. Vondrák was appointed as an Associate Professor at Stanford University in January 2016.

It was in 2006 that Mirzakhani began a collaboration with Alex Eskin of the University of Chicago. They tackled one of the hardest problems in their area, to generalise a result which Curtis McMullen had published in 2003 [11]:-

Mirzakhani, together with Alex Eskin and, in part, Amir Mohammadi, made a major breakthrough in understanding another dynamical system on moduli space that is related to the behaviour of geodesics in moduli space. Non-closed geodesics in moduli space are very erratic and even pathological, and it is hard to obtain any understanding of their structure and how they change when perturbed slightly. However, Mirzakhani et al have proved that complex geodesics and their closures in moduli space are in fact surprisingly regular, rather than irregular or fractal. It turns out that, while complex geodesics are transcendental objects defined in terms of analysis and differential geometry, their closures are algebraic objects defined in terms of polynomials and therefore have certain rigidity properties.

Eskin spoke about Mirzakhani, saying [9]:-

She is very optimistic, and that's infectious. When you work with her, you feel you have a much better chance of solving problems that at first seem hopeless.

Sometimes Eskin was pessimistic but Mirzakhani was not:-

Sometimes there were setbacks, but she never panicked.

After they were successful, she said:-

If we knew things would be so complicated, I think we would have given up. I don't know; actually, I don't know, I don't give up easily.

Eskin and Mirzakhani published Counting closed geodesics in moduli space in 2011.

In 2014 Mirzakhani became the first woman to be awarded a Fields Medal. This Medal was presented to her by the International Mathematical Union on 13 August 2014 at the International Congress of Mathematicians, held in Seoul, South Korea. The citation states that the award was:-

... for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

Mirzakhani won many honours during her short lifetime in addition to those we have already mentioned. She was awarded the Ruth Lyttle Satter Prize in Mathematics in San Diego on 10 January 2013. She was elected to the Paris Academy of Sciences in 2015, the American Philosophical Society in 2015, the National Academy of Sciences in 2016, and the American Academy of Arts and Sciences in 2017. She was an invited speaker at the Topology and Dynamical Systems & ODE section of the International Congress of Mathematicians in 2010 when she gave the lecture On Weil-Petersson volumes and geometry of random hyperbolic surfaces. Again at the International Congress of Mathematicians in Seoul, South Korea, in 2014 she was a plenary speaker.

Even before she was awarded the Fields Medal in 2014, Mirzakhani had been diagnosed with breast cancer. She continued working on mathematics producing not only results of great significance but developing tools along the way that will be used by researchers in the field as they continue to push forward. The cancer spread to her liver and bones and her death in a California hospital in July 2017 robbed mathematics of one of its brightest stars who, at the age of 40, was at the peak of her creativity. Stanford President Marc Tessier-Lavigne said following her death [12]:-

Maryam is gone far too soon, but her impact will live on for the thousands of women she inspired to pursue math and science. Maryam was a brilliant mathematical theorist, and also a humble person who accepted honours only with the hope that it might encourage others to follow her path. Her contributions as both a scholar and a role model are significant and enduring, and she will be dearly missed here at Stanford and around the world.

Ralph L Cohen, the Barbara Kimball Browning Professor of Mathematics at Stanford, said [12]:-

Maryam was a wonderful colleague. She not only was a brilliant and fearless researcher, but she was also a great teacher and terrific PhD adviser. Maryam embodied what being a mathematician or scientist is all about: the attempt to solve a problem that hadn't been solved before, or to understand something that hadn't been understood before. This is driven by a deep intellectual curiosity, and there is great joy and satisfaction with every bit of success. Maryam had one of the great intellects of our time, and she was a wonderful person. She will be tremendously missed.