陶哲轩Terence Tao

陶哲轩被朋友和同事称为Terry 陶哲轩。他的父亲雅克·德·比伊 陶哲轩是一位出生于中国的儿科医生，曾从事天才儿童教育和自闭症研究。Terry的母亲Grace出生于香港，拥有物理学和数学大学学位。德·比伊和Grace在香港大学学习时相识，并于1972年移民澳大利亚。Grace Tao在移民澳大利亚之前曾在香港多所中学教授物理、化学、科学和数学，到达澳大利亚后也在那里的中学任教。本传记的主人公Terry是他们的长子，有两个弟弟Trevor（比Terry小两岁）和Nigel（比Terry小四岁）。

Trevor和Nigel都极具天赋，他们后来获得博士学位并从事研究工作。Trevor 陶哲轩（生于1977年11月22日）从两岁起被诊断为自闭症。他在阿德莱德大学攻读双学位，理学学士（数学和计算机科学）与音乐学士（钢琴表演和作曲双主修），于2000年完成。他是国际象棋大师，最高国际象棋等级分2440，青少年时期还获得过数学和音乐（古典音乐作曲）奖项。Nigel于2000年获得澳大利亚国立大学计算机科学荣誉学士学位。他曾在匹兹堡的WhizBang! Labs工作，但该公司于2002年5月关闭。2006年，他开始在悉尼的谷歌工作。让我们回到Terry 陶哲轩的传记。

Terry两岁时，他的父母意识到他与其他孩子不同。他们看到他教五岁的孩子拼写和加法，当他们问他如何学会这些技能时，他回答说他在电视上看《芝麻街》。他三岁半时，父母送他去一所私立学校，但六周后，他们意识到他还没有准备好上学，而且老师也不知道如何教像他这样的孩子。于是他们让他退学，直到他像其他孩子一样五岁时才重新开始上学。文章[21]对Terry八岁生日前夕的数学能力进行了评估，当时他正在阿德莱德的Blackwood高中就读。Ken Clements写道，当他走进他家时，Terry正：-

……坐在房间的远角，读着一本名为《微积分》的精装书。陶哲轩个子很小，即使对一个七岁的孩子来说也是如此。见过他的两个弟弟后，陶哲轩陪我去了他父亲的书房，在那里，简短交谈后，我开始了我对异常聪慧的小学学龄儿童的常规评估程序。

Clements发现Terry知道群的定义，并能用微分微积分解决图形绘制问题。他想知道他的母亲教了他多少，但发现她的角色[21]：-

……更多是引导和激发陶哲轩的发展，而不是教他。她说陶哲轩喜欢自己阅读数学，他经常在放学后花三四个小时阅读数学教科书。

1984年，陶哲轩九岁时，他的父亲带他去拜访一些世界顶尖的数学家。他们去了普林斯顿的高等爱德华·斯图迪研究所，在那里陶哲轩接受了查尔斯·费夫曼和恩里科·邦别里的采访。查尔斯·费夫曼向陶哲轩提出了以下问题[8]：-

想象你被困在一个房间里，房间里有一头饥饿的狮子。你和狮子都被表示为空间中的点。假设狮子跑得比你快。假设你跑得比狮子快。假设你和狮子跑得完全一样快。你如何避免被吃掉？

尽管现在没有人记得陶哲轩的回答，但查尔斯·费夫曼说[8]：-

一个9岁的孩子能对一个数学问题提出想法，而这不是他在任何课堂上学到的常规内容，这让我印象深刻。

到Terry十一岁时，他的时间分配在1983年至1988年就读的Blackwood High School的学习和在阿德莱德的Flinders University上课之间，在那里他由Garth Gaudry（1941-2012）授课。甚至更早，在十岁时，他就开始参加国际数学奥林匹克。他在1986年获得铜牌，1987年获得银牌，1988年获得金牌，成为数学奥林匹克史上最年轻的金牌得主。但我们不应该认为每门学校科目对陶哲轩来说都很容易。他在有精确规则的科目上很出色，比如数学和拉丁语，但当他的英语老师要求全班写一篇关于“家”的文章时，他完全不知所措。

1989年，十四岁时，他开始在Flinders University进行全日制大学学习，并在1990年，当他还是一个15岁的学童时，写了他的第一本书Solving mathematical problems. A personal perspective。在书中，陶哲轩写道，一个问题的解决方案：-

……应当相对简短、易懂，并且最好带一点优雅。发现它应当是有趣的。

第一版于1992年出版，第二版于2006年出版；你可以在THIS LINK读到1990年的序言、2005年写的序言，以及2006年版书评的摘录。

他于1991年12月获得荣誉理学士学位。他说[109]：-

我仍然记得在澳大利亚弗林德斯大学读大学时的那次领悟：数学不仅仅是一场抽象符号的游戏，还可以用作分析和理解现代世界的工具。

他在[89]中写到了这一领悟的时刻：-

为什么有些统计数据可信，有些却不可信？为什么有些投资策略稳健，另一些却有风险？为什么计算机能在几秒钟内为你搜索整个互联网，却连一个哪怕只是稍微变形或模糊的印刷字都读不出来？为什么我们现代的卫星阵列能以惊人的精度告诉数百万司机他们的位置，却无法正确预测两周后的天气？数学知识可以回答这些问题，并使世界变得可理解、有秩序，而不是神秘莫测、反复无常。一旦我看到这种知识的力量，以及当一切“豁然贯通”、一个令人困惑的问题自行化为清晰解答时那种满足感，我就终生着迷了。我想用数学尽可能多地探索和理解这个世界。

然而，他本科阶段并非事事顺利[116]：-

在大学里，我有一门Fortran课，我抗拒它，理由是我已经会BASIC了；我记得我挑衅地用BASIC而不是Fortran写了期末作业代码（结果这门课挂了）。

他继续在弗林德斯大学攻读硕士学位，导师是Garth Gaudry。陶哲轩在[88]中写道：-

我从在弗林德斯大学读本科时就认识Garth了。他当时是院长（大致相当于美国的系主任），但仍能每周抽出一小时与我会面讨论实分析，当时我正在研读沃尔特·鲁丁的《实分析与复分析》，然后是埃利亚斯·施泰因的《奇异积分》，最终在他的指导下完成了一篇关于威廉·金顿·克利福德值奇异积分的硕士论文。

陶哲轩于1992年8月获得学位，撰写了学位论文Convolution operators generated by right-monogenic and harmonic kernels。他获得了弗林德斯大学颁发的大学奖章和富布赖特研究生奖学金，使他能够在美国进行研究。他在[109]中解释道：-

我其实并没有在四年内完成两个学位；我在正式入学前以非全日制身份就读于弗林德斯大学，他们决定将这些课程计入我的学位。

陶哲轩在1992年十六岁时前往美国开始读研究生[8]：-

研究生阶段是陶哲轩第一次离家，这一转变被证明是困难的。他的父亲在开学的第一周陪着他，教他如何做诸如洗衣服和开银行账户之类的基本事务。在空闲时间，陶哲轩加入了电影俱乐部，并玩桌上足球、在线桥牌和电脑游戏。

他开始在普林斯顿大学进行研究，导师是埃利亚斯·施泰因。起初事情很困难，因为在此之前陶哲轩一直觉得一切都很容易，以至于没有培养出任何学习技能。在一次考试中勉强及格后，他决心更加努力。他在[90]中写道：-

我与埃利亚斯·施泰因的每周会面往往是这样进行的：我会汇报过去一周我为解决当前研究问题所尝试的许多不同方法，但都没有取得多大成功；埃利亚斯·施泰因会耐心地听我说的一切，集中思考片刻，然后走到他的文件柜前，找出一篇预印本递给我，说：“我认为这篇论文的作者遇到了类似的问题，并用方法X解决了它。”然后我会回到办公室阅读这篇预印本，确实他们面临过类似的问题，我常常能借鉴其中的技巧来解决眼前的障碍（结果下周又会遇到新的障碍，但研究往往就是这样，尤其是作为研究生）。除其他事项外，这些会面让我深刻体会到数学经验的价值，因为能够在几分钟内在一个问题上取得比我整整一周所能完成的更关键的进展。

在他职业生涯的这个阶段，他在澳大利亚和美国两地度过时间。他是弗林德斯医学中心的助理研究员（1992-94），在他父亲的研究小组中担任计算机程序员，并在1993-94年期间担任普林斯顿大学的助理研究员。1995年，他获得了斯隆研究生奖学金。1996年6月，他凭借学位论文Three regularity results in harmonic analysis获得了博士学位。1996年，他的研究论文开始发表，当年发表了四篇论文。这些是：Weak-type endpoint bounds for Riesz means；（与Andrew C Millard合作）On the structure of projective group representations in quaternionic Hilbert space；On the almost everywhere convergence of wavelet summation methods；以及Convolution operators on Lipschitz graphs with harmonic kernels。

获得博士学位后，陶哲轩被任命为加州大学洛杉矶分校的Hedrick助理教授，他于1996年至1998年担任该职位。他继续担任加州大学洛杉矶分校的助理教授，并于2000年二十四岁时被提升为正教授。这使他成为加州大学洛杉矶分校有史以来最年轻的正教授。

当他开始在加州大学洛杉矶分校讲课时，他教的一个班上有一个名叫Laura的学生，她[114]：-

……她不记得在她开始把作业塞进教授办公室门缝后不久，是谁先提议他们去喝咖啡的。她喜欢陶哲轩那种放松而稳重的样子。

她后来毕业成为电气工程师，并在加州理工学院的美国宇航局喷气推进实验室工作。陶哲轩和Laura于2002年结婚，他们有两个孩子，William Tao和Madeleine 陶哲轩。当文章[114]写成时（2015年3月），William Tao十一岁，Madeleine 陶哲轩三岁。十一岁的William是[114]：-

……学习钢琴、吉他和单簧管。有一天，他在一家商场被一家人才中介公司看中——“我们喜欢他的脸，”他们说——他为莱斯特·佛特和迪士尼拍过广告。“他在某些方面和我不同，”陶哲轩说，他喜欢和儿子一起看《神秘博士》。“他喜欢数学，而且擅长数学，但他喜欢自由形式；他喜欢我从未涉足的东西，比如写作。我在学校从来都不太擅长……人文学科……任何更取决于观点的事情。”

2007年，陶哲轩被任命为加州大学洛杉矶分校的James and Carol Collins讲席教授。

为像陶哲轩这样正处于创造力巅峰的人写传记是非常困难的。任何关于他研究贡献的记述都会很快过时，因为他继续在极其广泛的不同领域中贡献重大成果。然而，他在年轻时就已经产出了一系列惊人的成果，从而获得了数学界所有最高奖项，而这位杰出的数学家仍在数学的多个领域不断取得重大进展。

在考察他的贡献之前，我们先列出他所获得的奖项和荣誉（尽管这份于2023年11月编制的名单，随着他不断获奖，注定会很快过时）。这些奖项包括：拉斐尔·塞勒姆奖（2000年）；American Mathematical Society的马克希莫·博谢纪念奖（2002年）；克莱数学研究所的克莱研究奖（2003年）；American Mathematical Society的Levi L Conant奖（2005年）；Australian Mathematical Society奖章（2005年）；国际分析、应用与计算学会的ISAAC奖（2005年）；SASTRA 拉马努金奖（2006年）；约翰·查尔斯·菲尔兹奖章（2006年）；亚历山大·奥斯特洛夫斯基基金会的亚历山大·奥斯特洛夫斯基奖（2007年）；美国国家科学基金会的Alan T Waterman奖（2008年）；昂萨格奖章（2008年）；信息论学会论文奖（2008年）；弗林德斯大学校友会毕业典礼奖（2008年）；费萨尔国王国际奖（数学）（2010年）；西北大学的Nemmers数学奖（2010年）；Society for Industrial and Applied Mathematics的乔治·波利亚奖（2010年）；克拉福德奖（2012年）；数学突破奖（2014年）；Royal Society of London的物理科学皇家奖章；PROSE奖“数学”类别（2015年）；波恩哈德·黎曼奖（2019年）；阿斯图里亚斯公主技术与科学研究奖（2020年）；Hungarian Academy of Sciences的鲍耶国际数学奖（2020年）；联合西格玛智能协会数学奖（2021年）；Advance Global Australian Awards的教育与研究奖（2022年）；全球澳大利亚年度人物奖（2022年），以及法国科学院的大奖章（2022年）。

关于授予陶哲轩这些奖项的更多信息，见THIS LINK。

此外，他还获得了斯隆基金会研究奖学金（1999-2001年）、弗洛伦斯·南丁格尔·大卫与露西尔·帕卡德基金会基金会奖学金（1999-2006年）、麦克阿瑟基金会的麦克阿瑟奖学金（2007-11年），并被任命为西蒙斯研究员（2012年）。他先后当选为澳大利亚科学院（2006年）、Royal Society会士（2007年）、National Academy of Sciences（2008年）和美国艺术与科学院（2009年）。他于2007年入围澳大利亚年度人物。

为了了解他的研究贡献，我们首先注意到他获得了2006年约翰·查尔斯·菲尔兹奖章：-

……表彰他对偏微分方程、组合学、调和分析和加性数论的贡献。

文章[1]描述了约翰·查尔斯·菲尔兹奖章的颁发，给出了如下概述：-

陶哲轩是一位顶尖的问题解决者，其卓越的工作影响了多个数学领域。他兼具纯粹的技术力量、灵光乍现的独创性以及令人惊讶的自然视角，让其他数学家不禁感叹：“为什么之前没人想到这一点？”31岁的陶哲轩已经撰写了八十多篇研究论文，合作者超过三十人，他的兴趣横跨数学的广阔领域，包括调和分析、非线性偏微分方程和组合数学。“我在多个领域工作，但我并不认为它们是割裂的，”他在克莱数学研究所年度报告发表的一次采访中说。“我倾向于将数学视为一个统一的学科，尤其高兴能有机会参与同时涉及多个领域的项目。”

我们在此注意到，他现在已大大超过了2006年文章中提到的八十篇研究论文，MathSciNet记录了1996年至2023年间的353篇出版物。宣布将约翰·查尔斯·菲尔兹奖章授予陶哲轩的新闻稿列举了他在多个领域的成就，这些成就促成了这一最负盛名的数学奖项的颁发。首先，它描述了他在素数分布方面与Ben Green的合作。他们证明了一个非凡的结果：素数包含任意长度的等差数列。他于2006年4月3日星期一在蒙特利尔大学作了关于 Long Arithmetic Progressions in the Primes的André-Aisenstadt讲座。以下是他的摘要[4]：-

塞迈雷迪·安德烈的一个著名而困难的定理断言，整数集的任何正密度子集都包含任意长的等差数列；这个定理有四种不同的证明（图论、遍历论、约瑟夫·傅里叶分析以及超图论），每一种都极具影响力、重要且深刻。一段时间以来，人们一直猜想同样的结果对素数也成立（素数当然具有零密度）。我将讨论最近与Ben Green合作获得这一猜想的工作，方法是将素数视为具有正相对密度的几乎素数（素因子很少的数）的子集。关键在于，由于筛法技术，如Goldston和Yildirim最近的工作，几乎素数比素数本身更容易控制。要将塞迈雷迪·安德烈的定理“转移”到这种相对设定中，需要借鉴塞迈雷迪·安德烈定理所有四种已知证明的技术，尤其是遍历论证明。

然而，这一了不起的成就只是众多成就之一，还有更多值得一说。陶哲轩做出许多贡献的一个领域是挂谷问题。这个问题最初在二维中提出，要求的是能够将一根针旋转180°的形状的最小面积。答案相当令人惊讶，事实上你可以使面积小于任何选定的数。陶哲轩研究了$n$维挂谷问题，其中最小体积同样可以任意小，但形状的分形维数未知。这个问题听起来相当专门，但恰恰相反，它与约瑟夫·傅里叶分析和非线性波有着惊人的联系。陶哲轩工作过的另一个领域是求解描述引力的广义相对论方程的特殊情形。对方程施加柱对称性会导致“波映射”问题，尽管该问题尚未解决，但陶哲轩的贡献引发了兴趣的极大复苏，因为他的想法似乎使解决方案成为可能。陶哲轩引入新颖想法、使该主题焕然一新的另一个领域是非线性埃尔温·薛定谔方程理论。这些方程有相当大的实际应用，陶哲轩的洞见再次对特定埃尔温·薛定谔方程的行为提供了大量启示。

让我们简要提及陶哲轩的其他一些显著贡献。2015年，他解决了埃尔德什差异问题，证明了埃尔德什在1932年提出的猜想。2016年，他提出了一种处理描述流体流动的克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程的新方法。关于这些方程有一些重要问题，它们出现在克莱数学研究所的七个千禧年问题中。陶哲轩继续研究这些方程，并在2019年取得了进一步的重大进展。同样在2019年，陶哲轩在解决洛萨·考拉兹猜想方面取得了进展。给定任意整数$n_{i}$，如果$n_{i}$是偶数，定义$n_{i+1} = \large\frac{1}{2}\normalsize n_{i}$，如果$n_i$是奇数，定义$n_{i+1} = 3n_{i} + 1$。洛萨·考拉兹在1937年提出的洛萨·考拉兹猜想指出，给定任意整数$n_{0}$，上述过程给出的序列总是在有限步内达到1 [36]：-

数学家们把洛萨·考拉兹猜想视为泥潭，互相警告要远离它。但现在陶哲轩取得的进展比任何人几十年来的都多。

陶哲轩的部分结果，该猜想仍未解决。

2022年，陶哲轩与Rachel Greenfeld一起找到了一个铺砌猜想的反例。2016年，Siddhartha Bhattacharya证明了如果只允许对瓷砖进行平移，那么单块瓷砖无法非周期性地铺满平面而不周期性地铺满平面。人们曾猜想这在更高维度中也成立，但陶哲轩和Greenfeld找到了一块充满扭曲和孔洞的单块瓷砖，它存在于一个非常高维的空间中（或许维度大于$2^{100^{100}}$），该瓷砖能非周期性地铺满那个空间但不能周期性地铺满。

陶哲轩与Tamar Ziegler合作在最近（2023年）的一篇论文中证明了以下定理[91]：-

... there exist infinite sets $A = \{ a_{1}, a_{2}, ... \}$ and $B = \{b_{1}, b_{2}, ... \}$ of natural numbers such that $a_{i} + b_{j}$ is prime whenever $1 ≤ i < j$。

人们可能会想，凭借其惊人的研究论文产出，陶哲轩不会有时间写书。然而，这完全错了，因为他既出版了研究专著，也出版了本科生教材。现在让我们来看看这些。2006年，陶哲轩出版了我们在上面提到的Solving mathematical problems。同年，他出版了一部两卷本教材Analysis。

关于这本书的更多信息，见THIS LINK。

同样在2006年，陶哲轩出版了Nonlinear dispersive equations，但令人惊讶的是，这仍然没有列完陶哲轩2006年的书，因为在那一年，他与Van Vu合作出版了Additive combinatorics。

关于这两本书的更多信息，见THIS LINK。

毫不奇怪，陶哲轩在他所做的一切事情上都是如此创新，他创造了一种新的书籍风格。上面描述的教材和研究专著在方法上具有创新性，但属于传统类型的书籍。2008年，陶哲轩出版了Structure and randomness. Pages from year one of a mathematical blog一书，2009年又出版了两本类似的书Poincaré's legacies, pages from year two of a mathematical blog第一部分和第二部分。你可以在THIS LINK给出的序言摘录中读到陶哲轩自己对这些博客的想法。

2010年，陶哲轩系列中的下一本An epsilon of room, I: real analysis. Pages from year three of a mathematical blog出版了。当我们在2011年撰写陶哲轩传记的第一版时，我们预期接下来几年会出现一系列漫长而引人入胜的著作。确实，我们是对的！他出版了An introduction to measure theory（2011年）、Topics in random matrix theory（2012年）、Higher order Fourier analysis（2012年）、Compactness and contradiction（2013年）、Hilbert's fifth problem and related topics（2014年）和Expansion in finite simple groups of Lie type（2015年）。我们注意到Hilbert's fifth problem and related topics获得了2015年PROSE奖最佳数学图书奖。关于这些书的详细信息，包括陶哲轩的序言和评论摘录，可在THIS LINK找到。

在访谈[116]中被问及爱好时，他回答说：-

如今，工作和家庭让我没有太多时间娱乐——基本上仅限于能在手机上几分钟内完成的活动。例如，近年来我每天花几分钟通过一个应用学习一门语言（目前是希伯来语）。……我会弹一点钢琴，尽管在这一点上，我认为我的两个孩子都超过了我。

让我们以来自8的一句引文结束：-

陶哲轩说，他的洞见是在大量艰苦工作之后才出现的。他从阅读、从其他数学家、从长途散步中获得想法。有时，一个想法会让他想起他在别处见过的类似问题，而那个问题现在可能有用。大多数路径都通向死胡同，但即使在死胡同里，他也能学到一些东西。“一旦你解决了一个问题，”他继续说道，“你往往只记得从A到B的那条短路径。你忘记了所有的死胡同。这有点遗憾。它给人一种错误的印象，即擅长数学的人只选择正确的步骤。但实际上有大量的试错和非常令人尴尬的想法。有时会有‘尤里卡’时刻，但更多时候是一种拍脑袋的时刻：‘当然，我为什么这么笨？’

Terence Tao is known to his friends and colleagues as Terry Tao. His father, Billy Tao, is a Chinese-born paediatrician who has undertaken research on educating gifted children and on autism. Terry's mother, Grace, was born in Hong Kong and has a university degree in physics and mathematics. Billy and Grace met while they were studying at the University of Hong Kong and they emigrated to Australia in 1972. Grace Tao taught physics, chemistry, science and mathematics in various secondary schools in Hong Kong before she emigrated to Australia and, once in Australia, also taught in secondary schools there. Terry, the subject of this biography, is their eldest child, having two younger brothers Trevor (two years younger than Terry) and Nigel (four years younger than Terry).

Both Trevor and Nigel are extremely talented, and they went on to be awarded Ph.D.'s and have research careers. Trevor Tao (born 22 November 1977) was diagnosed from age two as autistic. He studied at the University of Adelaide for a double degree, B.Sc. (Mathematics and Computer Science) with B.Mus. (double major in piano performance and composition), which he completed in 2000. He is an international chess master, highest chess rating 2440, who also won prizes in mathematics and music (classical music composition) when he was a teenager. Nigel was awarded a BSc (Hons) from the Australian National University in Computer Science in 2000. He worked for WhizBang! Labs in Pittsburgh, but they closed in May 2002. In 2006 he began working for Google in Sydney. Let us return to our biography of Terry Tao.

When Terry was two years old his parents realised that he was different from other children. They saw him teaching five year old children to spell and to add numbers and, when they asked him how he had learnt these skills, he replied that he had been watching Sesame Street on television. When he was three and a half years old his parents sent him to a private school but, six weeks later, they realised that he was not ready for schooling and also that the teachers did not know how to teach someone like him. So they removed him from the school and he did not start schooling again until he was, like other children, five years old. The article [21] gives an evaluation of Terry's mathematical abilities just before his eighth birthday by which time he was attending Blackwood High School, Adelaide. Ken Clements writes that when he went into his home, Terry was:-

... sitting in the far corner of a room reading a hardback book with the title 'Calculus'. Terence was small, even for a seven-year-old. After meeting his two brothers, I was accompanied by Terence to his father's study, where, after a brief chat, I began my usual assessment procedure for exceptionally bright primary school-age children.

Clements discovered that Terry knew the definition of a group and could solve graph sketching problems using the differential calculus. He wondered how much his mother was teaching him but found that her role [21]:-

... is more one of guiding and stimulating Terence's development than one of teaching him. She said that Terence likes to read mathematics by himself, and he often spent three or four hours after school reading mathematics textbooks.

In 1984, when Tao was nine years old, his father took him on a trip to meet some of the world's top mathematicians. They went to the Institute for Advanced Study at Princeton where Tao was interviewed by Charles Fefferman and Enrico Bombieri. Fefferman asked Tao the following question [8]:-

Imagine that you are trapped in a room with a hungry lion. Both you and the lion are represented as points in space. Suppose the lion can run faster than you. Suppose you can run faster than the lion. Suppose you and the lion can run at exactly the same speed. How do you avoid being eaten?

Although nobody can now remember Tao's answer, Fefferman said [8]:-

I was impressed that a 9-year-old kid could come up with ideas to a maths problem that wasn't a conventional thing he had learned in any class.

By the time Terry reached the age of eleven, he was dividing his time between his studies at Blackwood High School, which he attended from 1983 to 1988, and taking classes at Flinders University in Adelaide where he was taught by Garth Gaudry (1941-2012). Even earlier, at the age of ten, he began participating in International Mathematical Olympiads. He won a bronze medal in 1986, a silver medal in 1987 and a gold medal in 1988, becoming the youngest ever gold medalist in the Mathematical Olympiad. But we should not get the idea that every school subject was easy for Tao. He was brilliant at subjects with precise rules, like mathematics and Latin, but when his English teachers asked the class to write an essay about "home", he was totally stumped.

In 1989, at the age of fourteen he began full-time university studies at Flinders University and, in 1990 when a 15 year old schoolboy, wrote his first book Solving mathematical problems. A personal perspective. In it Tao writes that a solution to a problem:-

... should be relatively short, understandable, and hopefully have a touch of elegance. It should be fun to discover.

The first edition was published in 1992 with a second edition in 2006; you can read the 1990 Preface, the Preface written in 2005, and extracts from reviews of the 2006 edition at THIS LINK.

He was awarded a B.Sc. with Honours in December 1991. He said [109]:-

I still remember the realisation in college at Flinders University in Australia that mathematics was not just an abstract game of symbols but could be used as a tool to analyse and understand the modern world.

He wrote about this moment of realisation in [89]:-

Why are some statistics trustworthy and some not? Why are some investment strategies sound, and others risky? How come a computer can search the entire internet for you in a matter of seconds, but cannot read a printed word if it is even just slightly distorted or blurred? How come our modern array of satellites can tell millions of drivers their location with amazing accuracy, but cannot correctly predict the weather a fortnight into the future? Knowledge of mathematics can answer these questions, and make the world comprehensible and orderly rather than mysterious and capricious. Once I saw the power of this knowledge, and the satisfying feeling when everything "clicks" and one sees a confusing problem resolve itself into a clear solution, I was hooked for life. I wanted to use mathematics to explore and understand as much of the world as I could.

Not everything he did as an undergraduate went well, however [116]:-

At university, I had a Fortran class that I rebelled against, on the grounds that I already knew BASIC; I remember defiantly writing my final assignment code in BASIC instead of Fortran (and subsequently failing the class).

He continued to study at Flinders University for a Master's Degree advised by Garth Gaudry. Tao wrote in [88]:-

I knew Garth ever since I was an undergraduate at Flinders University. He was head of school then (a position roughly equivalent to department chair in the US), but still was able to spare an hour a week to meet with me to discuss real analysis, as I worked my way through Walter Rudin's "Real and complex analysis" and then Elias Stein's "Singular integrals", and then eventually completed a masters thesis under his supervision on Clifford-valued singular integrals.

Tao was awarded the degree in August 1992 having written the thesis Convolution operators generated by right-monogenic and harmonic kernels. He was awarded the University Medal by Flinders University and a Fulbright Postgraduate Scholarship to enable him to undertake research in the United States. He explained in [109]:-

I didn't really complete two degrees in four years; I attended Flinders part-time before my formal enrolment, and they decided to count those courses toward my degree.

Tao went to the United States in 1992 when he was sixteen years old to begin graduate school [8]:-

Graduate school was Tao's first time away from home, and the transition proved difficult. His father stayed with him for the first week of classes to teach his son how to perform such basic tasks as doing laundry and opening a bank account. In his free time, Tao joined the Film Club and played table football, online bridge, and computer games.

He began to undertake research at Princeton University advised by Elias Stein. At first things were difficult since Tao had found things so easy up until then that he had not developed any study skills. Scraping through an examination, he resolved to work harder. He writes in [90]:-

My weekly meetings with Eli would tend to go something like this: I would report on all the many different things I had tried over the past week, without much success, to solve my current research problem; Eli would listen patiently to everything I said, concentrate for a moment, and then go over to his filing cabinet and fish out a preprint to hand to me, saying, "I think the authors in this paper encountered similar problems and resolved it using Method X." I would then go back to my office and read the preprint, and indeed they had faced something similar and I could often adapt the techniques there to resolve my immediate obstacles (only to encounter further ones for the next week, but that's the way research tends to go, especially as a graduate student). Amongst other things, these meetings impressed upon me the value of mathematical experience, by being able to make more key progress on a problem in a handful of minutes than I was able to accomplish in a whole week.

At this stage in his career he spent time both in Australia and in the United States. He was an assistant researcher at Flinders Medical Centre (1992-94), working as a computer programmer in his father's research group, and also as an assistant researcher at Princeton University during 1993-94. In 1995 he was awarded a Sloan Postgraduate Fellowship. He was awarded his doctorate in June 1996 for his thesis Three regularity results in harmonic analysis. In 1996 his research papers began to appear in print, four papers being published in that year. These are: Weak-type endpoint bounds for Riesz means; (with Andrew C Millard) On the structure of projective group representations in quaternionic Hilbert space; On the almost everywhere convergence of wavelet summation methods; and Convolution operators on Lipschitz graphs with harmonic kernels.

Following the award of his doctorate, Tao was appointed Hedrick Assistant Professor at the University of California at Los Angeles, a position he held from 1996 to 1998. He continued as an assistant professor at the University of California at Los Angeles where, at the age of twenty-four, he was promoted to full professor in 2000. This made him the youngest person ever to be appointed as a full professor at UCLA.

When he began lecturing at UCLA, he taught a class containing a student named Laura who [114]:-

... doesn't remember who instigated the coffee date they went on some time after she started slipping homework under her professor's office door. She liked how relaxed and stable Tao seemed.

She went on to graduate as an electrical engineer and worked at NASA's Jet Propulsion Laboratory at the California Institute of Technology. Tao and Laura were married in 2002 and they have two children, William Tao and Madeleine Tao. When the article [114] was written (March 2015), William Tao was eleven years old and Madeleine Tao was three years old. William, at age eleven, was [114]:-

... learning piano, guitar and clarinet. One day he was scouted by a talent agency in a mall - "We like his face," they said - and he has appeared in commercials for Ford and Disney. "He's different from me in some ways," says Tao, who likes to watch Doctor Who with his son. "He likes maths, he's good at it, but he likes free-form; he likes things that I was never into, like writing. I was never very good at school with ... humanities ... anything which was more a matter of opinion."

In 2007 Tao was named the James and Carol Collins Professor at UCLA.

It is very difficult to write a biography of someone who is at the height of their creative powers as Tao is. Anything that one writes about his research contributions will be quickly outdated as he continues to contribute major results in a remarkably wide range of different areas. Yet at a young age, he produced a fantastic collection of results, leading to the award of all the top prizes in mathematics, and this outstanding mathematician continues to make major advances in diverse areas of mathematics.

Before looking at his contributions we note the prizes and awards he has received (although again this list, compiled in November 2023 is bound to become rapidly outdated as he continues to receive awards). These include: the Salem Prize (2000); the Bôcher Memorial Prize from the American Mathematical Society (2002); the Clay Research Award from the Clay Mathematical Institute (2003); the Levi L Conant Award from the American Mathematical Society (2005); the Australian Mathematical Society Medal (2005); the ISAAC Award from the International Society of Analysis, its Application and Computation (2005); the SASTRA Ramanujan Prize (2006); the Fields Medal (2006); the Ostrowski Prize from the Ostrowski Foundation (2007); the Alan T Waterman Award from the National Science Foundation (2008); the Onsager Medal (2008); the Information Theory Society Paper Award (2008); the Convocation Award from Flinders University Alumni Association (2008); the King Faisal International Prize (Mathematics) (2010); the Nemmers Prize in Mathematics from Northwestern University (2010); the George Polya Prize from the Society for Industrial and Applied Mathematics (2010); the Crafoord Prize (2012); the Breakthrough Prize in Mathematics (2014); the Royal Medal for physical sciences from the Royal Society of London; the PROSE award in the category of "Mathematics" (2015); the Riemann Prize (2019); the Princess of Asturias Award for Technical & Scientific Research (2020); the Janos Bolyai International Mathematical Prize by the Hungarian Academy of Sciences (2020); the United Sigma Intelligence Association Award for Mathematics (2021); the Education and Research Award by Advance Global Australian Awards (2022); Global Australian of the Year Award (2022), and the Grande Médaille of the French Academy of Sciences (2022).

For more information about these awards to Tao, see THIS LINK.

In addition he has received a Sloan Foundation research Fellowship (1999-2001), a Foundation Fellowship from the David and Lucille Packard Foundation (1999-2006), a MacArthur Fellowship from the MacArthur Foundation (2007-11), and appointed Simons Investigator (2012). He has been elected to the Australian Academy of Sciences (2006), to a fellowship of the Royal Society (2007), to the National Academy of Sciences (2008), and to the American Academy of Arts and Sciences (2009). He was a finalist in Australian of the Year in 2007.

To gain some insight into his research contributions, let us first note that he received the 2006 Fields medal:-

... for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.

The article [1], describing the award of the Fields Medal, gives this overview:-

Terence Tao is a supreme problem-solver whose spectacular work has had an impact across several mathematical areas. He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, "Why didn't anyone see that before?" At 31 years of age, Tao has written over eighty research papers, with over thirty collaborators, and his interests range over a wide swath of mathematics, including harmonic analysis, nonlinear partial differential equations, and combinatorics. " I work in a number of areas, but I don't view them as being disconnected," he said in an interview published in the Clay Mathematics Institute Annual Report. "I tend to view mathematics as a unified subject and am particularly happy when I get the opportunity to work on a project that involves several fields at once."

Let us note here that he has now greatly exceeded the eighty research papers mentioned in the 2006 article with MathSciNet recording a list of 353 publications between 1996 and 2023. The Press Release which announced the award of the Fields Medal to Tao listed his accomplishments in a number of areas which had led to the award of this most prestigious mathematical award. First it describes his work with Ben Green on the distribution of prime numbers. They proved the remarkable result that the primes contain arithmetic progressions of any length. He gave the André-Aisenstadt Lecture on Long Arithmetic Progressions in the Primes at the University of Montreal on Monday 3 April 2006. Here is his abstract [4]:-

A famous and difficult theorem of Szemerédi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the primes themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemerédi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemerédi's theorem, and especially from the ergodic theory proof.

This fantastic achievement, however, is only one of many, and there is so much more to say. An area to which Tao has made many contributions is that of the Kakeya problem. This problem, originally posed in two dimensions, asked for the minimum area of a shape in which one can rotate a needle through 180°. The answer is rather surprising, in fact you can make the area less than any chosen number. Tao has worked on the $n$-dimensional Kakeya problem where again the minimum volume can be made as small as one chooses, but the fractal dimension of the shape is unknown. This problem sounds rather specialised, but on the contrary there are surprising connections to Fourier analysis and nonlinear waves. Another area in which Tao has worked is solving special cases of the equations of general relativity describing gravity. Imposing cylindrical symmetry on the equations leads to the "wave maps" problem where, although it has yet to be solved, Tao's contributions have led to a great resurgence of interest since his ideas seem to have made a solution possible. Yet another area where Tao has introduced novel ideas, giving the subject a whole new look, is the theory of the nonlinear Schrödinger equations. These equations have considerable practical applications and again Tao's insights have shed considerable light on the behaviour of a particular Schrödinger equation.

Let us mention briefly some other remarkable contributions by Tao. In 2015 he solved the Erdős discrepancy problem, proving the conjecture made by Erdős in 1932. In 2016 he produced a new approach to the Navier-Stokes equations which describe fluid flow. There are important questions regarding these equations which figure in the Clay Mathematics Institute's seven Millennium Problems. Tao has continued working on these equations making further major progress in 2019. Also in 2019, Tao made progress in solving the Collatz conjecture. Given any integer $n_{i}$, then if $n_{i}$ is even, define $n_{i+1} = \large\frac{1}{2}\normalsize n_{i}$, if $n_i$ is odd, define $n_{i+1} = 3n_{i} + 1$. The Collatz conjecture, made by Lothar Collatz in 1937, states that given any integer $n_{0}$ the sequence given by the above procedure always reaches 1 in a finite number of steps [36]:-

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.

Despite Tao's partial result, the conjecture remains open.

In 2022, Tao, together with Rachel Greenfeld, found a counterexample to a tiling conjecture. In 2016 Siddhartha Bhattacharya proved that a single tile cannot tile the plane aperiodically but not periodically if only translations of the tile are allowed. It was conjectured that this would be true in higher dimensions, but Tao and Greenfeld found a single tile, full of twists and holes, which lives in a very high dimensional space (perhaps of dimension greater than $2^{100^{100}}$ ) which tiles that space aperiodically but not periodically.

A recent (2023) paper by Tao, in collaboration with Tamar Ziegler, proves the following theorem [91]:-

... there exist infinite sets $A = \{ a_{1}, a_{2}, ... \}$ and $B = \{b_{1}, b_{2}, ... \}$ of natural numbers such that $a_{i} + b_{j}$ is prime whenever $1 ≤ i < j$.

One might imagine that with his remarkable output of research papers, Tao would not find time to write books. However, this would be entirely wrong since he has produced both research monographs and undergraduate texts. Let us now look at these. In 2006 Tao published Solving mathematical problems which we mentioned above. In the same year he published a 2-volume textbook Analysis.

For more information about this book, see THIS LINK.

Also in 2006, Tao published Nonlinear dispersive equations, but, amazingly, this still does not complete the list of Tao's 2006 books for in that year, in collaboration with Van Vu, he published Additive combinatorics.

For more information about these two books, see THIS LINK.

It will come as no surprise to learn that Tao, who is such an innovator in everything he does, has created a new style of book. The textbook and research monographs described above are innovative in their approach but are traditional types of books. In 2008 Tao published the book Structure and randomness. Pages from year one of a mathematical blog and, in 2009, two similar books Poincaré's legacies, pages from year two of a mathematical blog Part I and Part II. You can read Tao's own thought on these blogs in the extracts from the Prefaces given at THIS LINK.

In 2010 the next in Tao's series was published An epsilon of room, I: real analysis. Pages from year three of a mathematical blog. When we wrote the first version of Tao's biography in 2011 we anticipated a long and fascinating series of books that would appear over the next years. Indeed, we were correct! He published An introduction to measure theory (2011), Topics in random matrix theory (2012), Higher order Fourier analysis (2012), Compactness and contradiction (2013), Hilbert's fifth problem and related topics (2014), and Expansion in finite simple groups of Lie type (2015). We note that Hilbert's fifth problem and related topics was awarded the 2015 Prose Award for Best Mathematics Book. There is detailed information about these books, including Tao's Prefaces and extracts from reviews, at THIS LINK.

When asked about hobbies in the interview [116], he replied:-

With work and family there isn't as much time for fun these days - pretty much restricted now to activities that can be done on my phone in a few minutes. For instance in recent years I spend a few minutes each day learning a language (currently Hebrew) through an app. ... I play a little piano, although at this point I think both my kids surpass me in this.

Let us end with a quote from [8]:-

Tao says that his insights arrive, when they do, after much hard work. He gets ideas from reading, from other mathematicians, from taking long walks. Sometimes, one idea reminds him of a similar problem he saw somewhere else that might prove useful now. Most paths lead nowhere, but he learns something even in the cul de sacs. "Once you solve a problem," he continues, "you tend to remember only the short path that got you from A to B. You forget all the dead ends. It's a bit of a shame. It gives the wrong impression that people who are good at mathematics only choose the right steps. But there is a lot of trial and error and really terribly embarrassing ideas. Sometimes there is a 'eureka' moment, but it's more of a hitting-your-head moment: 'Of course, why was I so dense?'