蒂莫西·高尔斯Timothy Gowers

蒂莫西·高尔斯被称为Tim。他的父母是Caroline Maurice和William Patrick Gowers，他有两个姐妹Rebecca 高尔斯和Katharine 高尔斯。Patrick Gowers是一位作曲家，以创作许多电影配乐而闻名，也以其吉他作品著称。他拥有剑桥大学博士学位：Eric Satie: his studies, notebooks and critics（1965年）。Rebecca 高尔斯是一位自由记者和作家。她的第一本书The Swamp of Death于2004年入围CWA非虚构类金匕首奖，最近她的书When to Walk入围了Orange Broadband小说奖长名单。Katharine 高尔斯是一位小提琴家，任职于：-

……非凡的音乐天赋和技术驾驭力。更重要的是她天生的风格感：一份无价的禀赋，将她的演奏提升到非凡成熟的水平。

事实上，高尔斯家族还有其他特别杰出的成员，例如Tim Gowers的曾祖父Ernest Arthur Gowers爵士 GCB GBE（1880-1966），他是一名公务员，但最著名的是在英语写作风格指南方面的工作。他编辑了Fowler的Modern English Usage，并写了一本名为Plain Words的书，该书仍在印刷中。

高尔斯被送到剑桥的国王学院学校，当时他是一名寄宿生，因为他的父母那时住在伦敦。鉴于我们已经提到的关于他姐姐凯瑟琳的情况，得知高尔斯极具音乐才能并且是该校的唱诗班歌手，就不会令人惊讶了。他在该校从玛丽·布里格斯那里接受了出色的数学教学，玛丽·布里格斯曾在格顿学院师从玛丽·卡特赖特。他获得了伊顿公学的国王奖学金
在那里[10]：-

我还有另一位鼓舞人心的老师，Norman Routledge，他曾是国王学院的会士。他不让自己局限于教学大纲，而是涉猎广泛得多。在伊顿的最后两年，数学专修生每周会拿到一张具有挑战性的问题单，这些问题即使有，也只是松散地基于教学大纲。当然，男孩子毕竟是男孩子，我们往往五天什么都不做，然后两天猛攻这些问题，但即便如此，这也是一次非常宝贵的经历。

在伊顿完成中学教育后，高尔斯进入剑桥大学三一学院。正是在他读本科期间，高尔斯决定自己想成为一名职业数学家[10]：-

我在读本科的某个时候确信自己想成为一名职业数学家——不过，即使在那时，我也几乎不知道这意味着什么。

然而，直到他在为数学荣誉学位考试第三部分学习期间选修了贝拉·波罗巴斯关于Banach spaces几何的课程，高尔斯才找到了一个他觉得自己适合开始研究的数学领域[10]：-

回想起来，当我做出如此重要的选择时，我对不同领域的研究会是什么样子几乎一无所知，这想起来很有趣。但我很幸运，发现自己身处一个非常适合我的领域，并且有一位出色的导师。

高尔斯于1988年与Emily Joanna Thomas结婚，她是Valerie Little和Keith Thomas爵士（历史学家、牛津大学基督圣体学院院长）的女儿；他们有两个儿子和一个女儿。1990年，高尔斯凭借其学位论文Symmetric Structures in Banach Spaces获得博士学位，贝拉·波罗巴斯是他的学位论文导师。他的第一篇论文Symmetric block bases in finite-dimensional normed spaces于1989年发表，同年他在奥地利施特罗布尔举行的“斯特凡·巴拿赫空间几何”会议上作了综述报告Symmetric sequences in finite-dimensional normed spaces。他于1989年被任命为三一学院研究员，担任此职位至1993年。他于1991年被任命为伦敦大学学院讲师，并在那里工作了四年。然而，从某种意义上说，他从未离开剑桥[10]：-

我过去常常从剑桥通勤，发现火车是一个适宜工作的场所，至少在上面取得过一次真正的突破。

他于1994年晋升为高级讲师，同年受邀在苏黎世举行的国际数学家大会上作报告，报告题目为Recent results in the theory of infinite-dimensional Banach spaces。在伦敦大学学院工作的四年里，他继续研究斯特凡·巴拿赫空间，并因此项工作于1995年被伦敦数学会授予初级Whitehead奖。该奖项的颁奖词写道[12]：-

伦敦大学学院的高尔斯因其将无穷组合学应用于解决斯特凡·巴拿赫空间理论中一系列长期存在的问题而获得初级Whitehead奖，其中一些问题最初由斯特凡·巴拿赫本人提出。高尔斯博士的成就包括：解决了著名的斯特凡·巴拿赫超平面问题（寻找一个不与任何超平面同构的斯特凡·巴拿赫空间），给出了斯特凡·巴拿赫空间恩斯特·施勒德-费利克斯·伯恩斯坦定理的一个反例，证明了如果一个斯特凡·巴拿赫空间的所有闭无穷维子空间都同构，则它是一个大卫·希尔伯特空间，以及给出了一个斯特凡·巴拿赫空间的例子，使得每个有界算子都是埃里克·伊瓦尔·弗雷德霍姆算子。在过去五年中，高尔斯使斯特凡·巴拿赫空间的几何面貌完全改观。他使用的技巧极具个人特色；特别是，他利用线性空间的弗兰克·普伦普顿·拉姆齐理论，陈述子空间而非子序列的二分法。在这个最初结构很少的领域，需要高水平的想象力和技术实力。这项工作展示了这两个特点，这些技巧似乎很可能在未来找到在不同领域的进一步应用。

1995年，高尔斯被任命为剑桥大学讲师。次年，他在匈牙利布达佩斯举行的第二届欧洲数学大会上被授予欧洲数学会奖。该奖项的颁奖词如下：-

高尔斯的工作使斯特凡·巴拿赫空间的几何面貌完全改观。仅举其若干引人注目的结果：他解决了著名的斯特凡·巴拿赫超平面问题，即找到一个不同构于其任何超平面的斯特凡·巴拿赫空间。他给出了斯特凡·巴拿赫空间上恩斯特·施勒德-费利克斯·伯恩斯坦定理的一个反例。他证明了斯特凡·巴拿赫空间的一个深刻二分原理，若与Komorowski和Tomczak-Jaegermann的结果相结合，则表明：如果一个斯特凡·巴拿赫空间的所有闭无限维子空间都同构于该空间，那么它就是一个希尔伯特空间。他（与Maurey合作）给出了一个斯特凡·巴拿赫空间的例子，使得从该空间到自身的每个有界算子都是埃里克·伊瓦尔·弗雷德霍姆算子。他的数学既极具原创性，又在技术上非常强。他所使用的技巧高度个性化；特别是，他非常巧妙地运用了无限弗兰克·普伦普顿·拉姆齐理论。

在欧洲数学大会上，高尔斯作了关于Banach spaces with few operators的报告。两年后，他在1998年于柏林举行的国际数学家大会上获得了约翰·查尔斯·菲尔兹奖。颁奖词开头写道[4]：-

高尔斯对分析做出了重要贡献，广泛运用了组合理论的方法。这两个领域表面上几乎没有关系，而高尔斯的一项重要成就就是卓有成效地将它们结合起来。

颁奖词结尾写道：-

一年前，高尔斯在组合分析领域引起了关注，当时他为数学家塞迈雷迪·安德烈的一个定理给出了一个新的证明，比原来的论证路线更简短、更优雅。这样的成就需要极其深刻的数学理解。

1998年，高尔斯被任命为剑桥大学Rouse Ball数学教授。1999年，他成为伦敦皇家学会会士。他继续发表具有重大意义的论文，如Hypergraph regularity and the multidimensional Szemerédi theorem（2007）；Gabor Sarkozy的评论开头写道：-

在这篇突破性论文中，作者证明了他版本的超图正则引理及相关的计数引理。作为应用，他首次给出了哈里·弗斯腾伯格和Katznelson的多维塞迈雷迪·安德烈定理的组合证明，并且是第一个提供显式界的证明。

高尔斯另一篇近期极具重要性的论文是Quasirandom groups（2008），但我们将提及高尔斯的几项重要工作，这些工作除了他惊人的研究贡献外，也是对数学的重大贡献。首先让我们提及他的书Mathematics. A very short introduction（2002）。这本书包含八章：用数学来模拟现实世界意味着什么？；数字是什么，它们在什么意义上存在（尤其是“虚”数）？；什么是数学证明？；无限小数意味着什么，为什么这很微妙？；讨论高维（例如26维）空间意味着什么？；非欧几何是怎么回事？；数学如何解决那些无法精确回答、只能近似回答的问题？；数学家是否真的在25岁时就江郎才尽？以及关于数学界的其他社会学问题。

让我们以给出高尔斯参与的两个进一步创新项目的细节来结束这篇传记。第一个是The Princeton Companion to Mathematics （2008）一书，由高尔斯担任编辑，June 伊萨克·巴罗-Green和Imre Leader担任副编辑。陶哲轩开始一篇评论写道：-

《普林斯顿数学指南》是一部独特的文本，它并不完全属于数学写作的任何通常类别。它不完全是数学百科全书，不完全是数学综述集，不完全是数学的通俗导论，当然也不是数学教科书；然而它仍然是一部极其丰富和有价值的参考著作，涵盖了当今现代数学的几乎所有方面（尽管肯定侧重于研究层面的纯数学）。百科全书可能主要关注定义，综述文章可能关注历史或最新研究，通俗导论可能关注类比、人物或娱乐性叙述；相比之下，这本书旨在回答（或至少涉及）关于数学的基本问题，如“什么是算术几何？”，“我们为什么关心函数空间？”，“数学今天如何在生物学中使用？”，“波恩哈德·黎曼假设的意义是什么？”，“为什么有这么多数系？”或“数学研究就是严格证明定理吗？”。

陶哲轩以这样的话结束他的评论：-

总之，这本独特的书是一部范围极其广泛、出人意料地易于理解的参考著作，涵盖了现代（以及历史上的）数学中相当大的一部分。虽然它绝不能替代更传统的数学教科书，但它与那些更详细、更精确、更技术性的文本形成了很好的互补，并且是少数几个能让人真正把全部数学看作一个统一学科的地方之一，其中有着连贯的主题和目标。

我们提到的最后一个项目是“Polymath项目”。高尔斯建议：-

……如果一大群数学家能够高效地连接他们的大脑，他们或许也能非常高效地解决问题。

Michael Nielsen解释说：-

运用与开源编程项目类似的原则，[高尔斯]利用博客和wiki组织了一次开放的数学合作，试图为一个称为密度Hales-Jewett定理的重要数学定理找到新的证明。

这个项目非常成功，尽管参与的数学家也许比高尔斯所希望的少。

高尔斯的第一段婚姻于2007年解除，2008年他与Julie Barrau结婚；他们有一个儿子。

高尔斯在2012年女王生日授勋名单中被授予爵士爵位。2012年6月16日宣布，爵士爵位授予剑桥大学纯数学与数理统计系皇家学会研究教授、皇家学会会士William 高尔斯教授：表彰其对数学的贡献。2013年他获得了进一步的荣誉，9月13日星期五，在圣安德鲁斯Younger Hall举行的一场特别毕业典礼上，他被授予荣誉学位，该典礼是圣安德鲁斯大学600周年庆典的一部分。他是十七位“国际学者和思想家”、“我们这一代最杰出的一些头脑”之一，以这种方式受到表彰。

圣安德鲁斯大学数学与统计学院的Kenneth Falconer所作的颂词见THIS LINK

最后让我们引用高尔斯关于数学的一些话：-

用非常概括的话来说，我想如果你把数学分为使用“初等”方法的数学和使用大量复杂理论及成熟技巧的数学，那么我更倾向于前者而非后者。

我只是尝试找到某些我可能能够有效思考的问题，而且很多时候我思考问题却毫无进展。……对于我思考的大多数问题，我都毫无进展，但对于斯特凡·巴拿赫空间和组合数学，两者的情况都是，有些问题似乎值得解决，我发现它们很有趣。

我喜欢谈论配方法，当你有一个结果A，它在一个方向上推广为结果B，在另一个方向上推广为结果C：那么你想找到C的推广，它对应于B如何推广A。

我可以在家里和办公室工作，那是我最常工作的两个地方。只要有任何地方，我有一叠纸和一支圆珠笔……但如果你坐在机场休息室里等待，对许多人来说这会是非常无聊的经历，对数学家来说却不是。拿出一些纸来思考一些问题。

我们鼓励读者进一步查看高尔斯在[6]、[7]和[8]中阐述的关于数学的想法。

Timothy Gowers is known as Tim. His parents are Caroline Maurice and William Patrick Gowers, and he has two sisters Rebecca Gowers and Katharine Gowers. Patrick Gowers is a composer, famous for composing the music for many films and also known for his works for the guitar. He has a doctorate from the University of Cambridge: Eric Satie: his studies, notebooks and critics (1965). Rebecca Gowers is a freelance journalist and author. Her first book, The Swamp of Death, was shortlisted in 2004 for a CWA Golden Dagger Award for Non-Fiction and, more recently, her book When to Walk was longlisted for the Orange Broadband Prize for Fiction. Katharine Gowers is a violinist with:-

... exceptional gifts of musicianship and technical command. Even more important is her innate sense of style: a priceless gift which lifts her playing to a level of exceptional maturity.

In fact the Gowers family has other exceptionally distinguished members such as Tim Gowers' great-grandfather Sir Ernest Arthur Gowers GCB GBE (1880-1966) who was a civil servant, but best known for work on style guides for writing the English language. He edited Fowler's Modern English Usage, and wrote a book titled Plain Words which is still in print.

Tim Gowers was sent to King's College School, Cambridge where he was a boarder as his parents were living in London at the time. Given what we have already mentioned about his sister Katharine, it will come as no surprise to learn that Gowers is extremely musical and was a chorister at the School. He had some excellent mathematics teaching at the School from Mary Briggs, who had studied under Mary Cartwright at Girton College. He won a King's scholarship to Eton College
where [10]:-

I had another inspirational teacher, Norman Routledge, who had been a fellow of King's. He did not allow himself to be limited to the syllabus but ranged far more widely. In my last two years at Eton, the mathematics specialists were given a weekly sheet of challenging problems which were only loosely based on the syllabus, if at all. Of course, boys being boys, we tended to do nothing for five days and then rush at them for two days, but even so it was a very valuable experience.

After completing his school education at Eton, Gowers matriculated at Trinity College, Cambridge. It was while he was an undergraduate that Gowers decided that he wanted to become a professional mathematician [10]:-

I became certain that I would like to be a professional mathematician some time when I was an undergraduate - though, even then, I had little idea of what this meant.

However, it was not until he took Béla Bollobás's course on the geometry of Banach spaces while studying for Part III of the Mathematical Tripos that Gowers found an area of mathematics which he felt was the right one for him to begin research [10]:-

Looking back it is amusing to remember how little idea I had of what research in different areas would be like when I made such an important choice. But I was lucky and found myself in an area that suited me very well, and with an excellent supervisor.

Gowers married Emily Joanna Thomas, daughter of Valerie Little and Sir Keith Thomas (historian and President of Corpus Christi College Oxford) in 1988; they had two sons and a daughter. In 1990 Gowers was awarded his doctorate for his thesis Symmetric Structures in Banach Spaces written with Béla Bollobás as his thesis advisor. His first paper Symmetric block bases in finite-dimensional normed spaces was published in 1989 and in the same year he gave a survey lecture Symmetric sequences in finite-dimensional normed spaces to the conference 'Geometry of Banach spaces' held in Strobl, Austria. He was appointed as a Research Fellow at Trinity College in 1989, holding this position until 1993. He was appointed as a Lecturer at University College, London, in 1991 and spent four years there. However, in some sense he never left Cambridge [10]:-

I used to commute from Cambridge, and found the train a congenial place to work, making at least one genuine breakthrough on it.

He was promoted to Reader in 1994 and in the same year was an invited speaker at the International Congress of Mathematicians held in Zürich where he gave the address Recent results in the theory of infinite-dimensional Banach spaces. During the four years he spent at University College he continued to work on Banach spaces and he was awarded the 1995 Junior Whitehead Prize by the London Mathematical Society for this work. The citation for the prize reads [12]:-

Dr W T Gowers of University College, London, is awarded a Junior Whitehead Prize for his work in applying infinite combinatorics to resolve a series of longstanding questions in Banach space theory, some originating with Banach himself. Dr Gowers' achievements include the following: a solution to the notorious Banach hyperplane problem (to find a Banach space which is not isomorphic to any hyperplane), a counterexample to the Banach space Schröder-Bernstein theorem, a proof that if all closed infinite-dimensional subspaces of a Banach space are isomorphic then it is a Hilbert space, and an example of a Banach space such that every bounded operator is a Fredholm operator. Over the past five years, Gowers has made the geometry of Banach spaces look completely different. The techniques he uses are highly individual; in particular, he makes use of a Ramsey theory for linear spaces, stating a dichotomy for subspaces rather than subsequences. In this area, where there is initially little structure, imagination and technical strength of a high calibre are needed. The work demonstrates both characteristics, and the techniques seem likely to find further application in different fields in the future.

In 1995 Gowers was appointed as a lecturer at the University of Cambridge. In the following year he was awarded a European Mathematical Society Prize at the 2nd European Congress of Mathematics held in Budapest, Hungary. The citation for the Prize reads:-

William Timothy Gowers' work has made the geometry of Banach spaces look completely different. To mention some of his spectacular results: he solved the notorious Banach hyperplane problem, to find a Banach space which is not isomorphic to any of its hyperplanes. He gave a counterexample to the Schröder-Bernstein theorem for Banach spaces. He proved a deep dichotomy principle for Banach spaces which, if combined with a result of Komorowski and Tomczak-Jaegermann, shows that if all closed infinite-dimensional subspaces of a Banach space are isomorphic to the space, then it is a Hilbert space. He gave (jointly with Maurey) an example of a Banach space such that every bounded operator from the space to itself is a Fredholm operator. His mathematics is both very original and technically very strong. The techniques he uses are highly individual; in particular, he makes very clever use of infinite Ramsey theory.

At the European Congress, Gowers lectured on Banach spaces with few operators. Two years later, he received a Fields Medal at the International Congress of Mathematicians held in Berlin in 1998. The citation begins [4]:-

William Timothy Gowers has provided important contributions to functional analysis, making extensive use of methods from combination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully.

The citation ends:-

A year ago, Gowers attracted attention in the field of combination analysis when he delivered a new proof for a theorem of the mathematician Emre Szemerédi which is shorter and more elegant than the original line of argument. Such a feat requires extremely deep mathematical understanding.

In 1998 Gowers was named Rouse Ball Professor of Mathematics at Cambridge. In 1999 he became a fellow of the Royal Society of London. He continues to produce papers of great significance such as Hypergraph regularity and the multidimensional Szemerédi theorem (2007); Gabor Sarkozy's review begins:-

In this breakthrough paper the author proves his version of the Hypergraph Regularity Lemma and the associated Counting Lemma. As an application, he gives the first combinatorial proof of the multidimensional Szemerédi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound.

Another highly significant recent paper by Gowers is Quasirandom groups (2008) but we will mention several important works by Gowers which are major contributions to mathematics in addition to his amazing research contributions. First let us mention his book Mathematics. A very short introduction (2002). The book contains eight chapters: What does it mean to use mathematics to model the real world?; What are numbers, and in what sense do they exist (especially "imaginary" numbers)?; What is a mathematical proof?; What do infinite decimals mean, and why is this subtle?; What does it mean to discuss high-dimensional (e.g. 26-dimensional) space?; What's the deal with non-Euclidean geometry?; How can mathematics address questions that cannot be answered exactly, but only approximately?; Is it true that mathematicians burn out at the age of 25? and other sociological questions about the mathematical community.

Let us end this biography with giving some details of two further innovative projects in which Gowers has been involved. The first of these is the book The Princeton Companion to Mathematics (2008) with Gowers as editor, and June Barrow-Green and Imre Leader as associate editors. Terence Tao begins a review writing:-

The Princeton companion to mathematics is a unique text, which does not fall neatly into any of the usual categories of mathematical writing. It is not quite a mathematical encyclopedia, it is not quite a collection of mathematical surveys, it is not quite a popular introduction to mathematics, and it is certainly not a mathematics textbook; and yet it is still an immensely rich and valuable reference work that covers almost all aspects of modern mathematics today (although there is certainly an emphasis on pure mathematics at the research level). An encyclopedia might focus primarily on definitions, a survey article might focus on history or on the latest research, and a popular introduction might focus on analogies, personalities or entertaining narrative; in contrast, this book is intended to answer (or at least address) basic questions about mathematics, such as "What is arithmetic geometry?", "Why do we care about function spaces?", "How is mathematics used today in biology?", "What is the significance of the Riemann hypothesis?", "Why are there so many number systems?'", or "Is mathematical research all about proving theorems rigorously?".

Tao ends his review by writing:-

In summary, this unique book is an extraordinarily broad and surprisingly accessible reference work for a remarkably large fraction of modern (and historical) mathematics. While it does not substitute by any means for more traditional textbooks in mathematics, it complements these more detailed, precise, and technical texts nicely, and is one of the rare places where one can actually see all of mathematics as a unified subject, with coherent themes and goals.

The final project we mention is the "Polymath project." Gowers suggested:-

... if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.

Michael Nielsen explains:-

Using principles similar to those employed in open source programming projects, [Gowers] used blogs and a wiki to organize an open mathematical collaboration attempting to find a new proof of an important mathematical theorem known as the density Hales-Jewett theorem.

The project has been very successful although perhaps fewer mathematicians took part than Gowers had hoped.

Gowers' first marriage was dissolved in 2007 and in 2008 he married Julie Barrau; they have one son.

Gowers was Knighted in the Queen's Birthday Honours List 2012. On 16 June 2012 it was announced that the Knighthood was to Professor William Timothy Gowers, FRS, Royal Society Research Professor, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge: For services to Mathematics. He received a further honour in 2013 when, on Friday 13 September, in the Younger Hall, St Andrews, he was given an honorary degree during a special graduation ceremony which formed part of the University of St Andrews' 600th Anniversary celebrations. He was one of seventeen "international scholars and thinkers", "some of the best minds of our generation", who were honoured in this way.

The laureation address by Kenneth Falconer, School of Mathematics and Statistics, University of St Andrews, is at THIS LINK

Finally let us quote some words by Gowers about mathematics:-

In very general terms I suppose if you divide mathematics into that which uses 'elementary' methods and mathematics that uses a lot of sophisticated theory and well-established techniques, then I'm drawn towards the former rather than the latter.

I just try and find certain problems that I might be able to think about profitably, and quite often I think about problems and get absolutely nowhere. ... with most problems I think about I get absolutely nowhere, but with Banach spaces and combinatorics it was just the case with both of them, there were problems that seemed reasonable to tackle, and I found them interesting.

I like to talk about completing the square, when you have result A that generalises in one direction to result B and in another direction to result C: then you want to find the generalisation of C that corresponds to how B generalises A.

I can work at home and in my office, and those are the two places I work most. Just anywhere where I've got a pad of paper and a biro ... But if you're sitting waiting in an airport lounge, which for many people would be a very boring experience, for a mathematician it isn't. Get out some paper and have a think about things.

We encourage the reader to look further at Gowers' ideas about mathematics set out in [6], [7] and [8].