理查·博赫兹Richard Borcherds

理查·博赫兹 的父亲 Peter Howard Borcherds 曾学习电气工程并获得学士学位，但后来他的兴趣转向了数学物理。在获得物理学硕士和博士学位后，Peter Borcherds 成为开普敦大学的物理学讲师。Peter 与 Margaret Elizabeth Greenfield 结婚。Peter 和 Margaret 有四个孩子；博赫兹 的三个兄弟中有两个后来成为数学教师，Michael Borcherds 作为数学教学软件包 GeoGebra 的首席开发者尤为知名。当 博赫兹 大约一岁时，他的父母离开南非前往英国的奥尔德马斯顿。Peter Borcherds 后来成为伯明翰大学的物理学讲师；Peter 的特别兴趣是计算机在物理学中的应用以及科学家参与政治。

博赫兹 就读于伯明翰的爱德华国王学校。他是一名热衷的国际象棋手，到十四岁时，他已成为中部地区 21 岁以下国际象棋冠军。童年时，他接触了相当多的数学，包括 哈罗德·斯科特·麦克唐纳·考克斯特 的多面体论文以及 Cundy 和 Rollett 的 Mathematical models。

博赫兹进入剑桥大学三一学院，在那里他是一名本科生。获得学士学位后，他继续在约翰·康威的指导下进行研究。起初他觉得自己不具备成为研究数学家的条件[8]：-

我没有取得很大进展。大部分时间我都在努力保住我的工作。我看到其他同龄人，比如西蒙·唐纳森（1986年Fields Medallist），要成功得多，我想我显然没有那么优秀。有几次我想过退学。

然而，他非常成功，并于1985年凭借其学位论文The Leech lattice and other lattices获得博士学位。John Leech于1965年发现的Leech格给出了24维空间中球体的密集堆积。从这个格的自同构群中，约翰·康威在1968年发现了三个此前未知的有限单群。博赫兹在其学位论文的前言中写道：-

我感谢我的研究导师约翰·康威教授的帮助和鼓励。我也感谢S.E.R.C.的财务支持以及三一学院的研究奖学金和会士职位。

他于1985年发表了一篇论文The Leech lattice，并在次年发表了Vertex algebras, Kac-Moody algebras and the monster。他发明了顶点代数的概念，这已被证明极其重要，但起初很少有人意识到其重要性[8]：-

当时我对此相当满意，但几年后我有点失望，因为很明显没有其他人真正对它感兴趣。有一个复杂到没人能理解的想法是没有意义的。我记得我过去常常做关于顶点代数的演讲，通常没有人来听。然后有一次我有了一个非常大的听众。但是有一个印刷错误，标题写的是“涡旋代数”，而不是“顶点代数”。听众是流体物理学家，当他们意识到这是一个印刷错误时，他们对我所说的也不感兴趣。

博赫兹于1983年被任命为剑桥大学三一学院的研究员，并担任此职位直到1987年。然后他在1987-88学年担任加州大学乔治·伯克利分校的莫里助理教授，之后回到剑桥，于1988年被任命为皇家学会大学研究员。他最著名的成果是在1989年春天证明了“月光猜想”。他花了八年时间思考这个猜想，同时产生了许多其他非常重要的结果。他在印度旅行时获得了证明这个猜想所需的灵感[8]：-

我当时在克什米尔。我一直在印度北部旅行，有一次非常漫长而烦人的巴士旅程，持续了大约24小时。然后巴士不得不停下来，因为发生了山体滑坡，我们无法继续前进。这一切都相当不愉快。总之，我只是在这次巴士旅程中玩弄一些计算，最终我找到了一个让一切奏效的想法。

月光猜想背后的想法很难解释。以下是Allyn Jackson在[5]中写的：-

在有限单群的分类中，发现的最神秘的物体之一是魔群。有各种猜想试图将魔群与数学的其他部分联系起来。博赫兹发明了顶点代数的概念，并用它来解决约翰·康威-Norton猜想，该猜想涉及魔群的表示论（这个理论有时被称为“魔群月光”）。他使用这些结果为某些模形式和自守形式生成了乘积公式。第一个这样的公式是在一维情况下由莱昂哈德·欧拉和卡尔·古斯塔夫·雅各布·雅可比发现的，而代数几何中的传统智慧是这种乘积公式不可能存在于更高维度。博赫兹的工作在物理学中也很重要，因为它为二维共形场论奠定了严格的基础。

博赫兹一直担任剑桥的皇家学会大学研究员直到1992年，然后在下一年担任剑桥的讲师。1993年，他被任命为加州大学伯克利分校的数学教授。他于1996年回到剑桥，在数学系担任皇家学会教授三年，然后于1999年回到加州大学伯克利分校的教授职位。他继续担任这个职位。

1992年，博赫兹被伦敦数学会授予初级Whitehead奖。颁奖词写道：-

1992年初级Whitehead奖授予剑桥大学的博赫兹博士，以表彰他在共形场论的数学方面所做的工作。博赫兹在数学上做出了许多重要而原创的贡献，其应用引人注目且出人意料。一项显著的工作是他将马克·卡茨-Moody代数推广，经典表示论的大部分内容都适用于此。对于他这一类别中的一个代数，赫尔曼·外尔分母公式的类似物给出了经典模函数作为无穷乘积的一个非凡且相当出人意料的分解。博赫兹发明的另一个重要概念是“顶点算子代数”。这是表述共形量子场论基本数学思想的一种方式。这一概念被雅科夫·弗仑克尔、Lepowski和Meurman在他们关于Monster单群的工作中广泛使用，并且最近它已成为E弗仑克尔在弗拉基米尔·德林费尔德关于环路群表示的罗伯特·朗兰兹对应猜想上取得进展的重要工具。在他一直在发展的思想的精彩综合中，博赫兹证明了约翰·康威和Norton关于Monster群的“月光”模的猜想。

同样在1992年，他在巴黎欧洲数学家大会上获得了欧洲数学会颁发的奖项。1994年3月10日，他当选为皇家学会会士。然而，他最大的荣誉是在1998年8月18日于柏林国际数学家大会开幕式上被授予约翰·查尔斯·菲尔兹奖章。颁奖词如下：-

……表彰他对代数、自守形式理论和数学物理的贡献，包括引入顶点代数和博赫兹的索菲斯·李代数，证明约翰·康威-Norton月光猜想以及发现一类新的自守无穷乘积。

博赫兹从容地对待伴随这一荣誉而来的名声[8]：-

获奖之前，我常认为它极其重要，但现在我意识到它毫无意义。然而，当我证明了月光猜想时，我欣喜若狂。如果我得到一个好结果，我会为此高兴好几天。我有时想知道这是否就是服用某些药物时的感觉。我实际上不知道，因为我还没有验证过我这个理论。

博赫兹的妻子是拓扑学家Ursula Gritsch，他们有两个女儿。他当前的研究兴趣包括量子场论的数学严格化方法。

Richard Borcherds's father, Peter Howard Borcherds, had studied electrical engineering for his B.Sc. but then his interests moved towards mathematical physics. After the award of an M.Sc. and Ph.D. in physics, Peter Borcherds became a lecturer in physics at the University of Cape Town. Peter had married Margaret Elizabeth Greenfield. Peter and Margaret had four children; two of Richard's three brothers went on to become mathematics teachers, Michael Borcherds being particularly well known as the lead developer of the mathematics teaching package GeoGebra. When Richard was about a year old his parents left South Africa and went to Aldermaston in England. Peter Borcherds went on to became a lecturer in physics at Birmingham University; Peter's particular interests are the use of computers in physics and the involvement of scientists in politics.

Richard attended King Edward's School in Birmingham. He was a keen chess-player and by the age of fourteen, he was Midlands under 21 Chess Champion. As a child he was exposed to quite a lot of mathematics, including Coxeter's polyhedron paper and Cundy and Rollett's Mathematical models.

Borcherds entered Trinity College, Cambridge where he was an undergraduate. After the award of his B.A., he proceeded to undertake research supervised by John Conway. At first he felt that he did not have what it takes to be a research mathematician [8]:-

I wasn't getting very far. Most of the time I was struggling to keep my job. I'd see other people my age, such as Simon Donaldson (1986 Fields Medallist), being considerably more successful, and I thought I'm obviously not all that good. There were times when I thought of dropping out.

However, he was highly successful and was awarded his doctorate in 1985 for his thesis The Leech lattice and other lattices. The Leech lattice, discovered by John Leech in 1965, gave a dense packing of spheres in 24 dimensions. From the automorphism group of this lattice, Conway had discovered three previously unknown finite simple groups in 1968. Borcherds wrote in the Preface to his thesis:-

I thank my research supervisor Professor J H Conway for his help and encouragement. I also thank the S.E.R.C. for its financial support and Trinity College for a research scholarship and a fellowship.

He published a paper The Leech lattice in 1985 and Vertex algebras, Kac-Moody algebras and the monster in the following year. He had invented to the idea of a vertex algebra which has proved to be extremely significant but at first few realised its importance [8]:-

I was pretty pleased with it at the time but after a few years I got a bit disillusioned, because it was obvious that nobody else was really interested in it. There is no point in having an idea that is so complicated that nobody can understand it. I remember I used to give talks on vertex algebras, and usually nobody turned up. Then there was this one time when I got a really big audience. But there had been a misprint, and the title read "vortex algebras", not "vertex algebras". The audience was made up of fluid physicists, and when they realised it was a misprint, they weren't interested either in what I had to say.

Borcherds had been appointed as a Research Fellow at Trinity College, Cambridge, in 1983 and held this post until 1987. He then spent the academic year 1987-88 as Morrey Assistant Professor at the University of California, Berkeley, before returning to Cambridge where he was appointed as a Royal Society University Research Fellow in 1988. His most famous result was to prove the "moonshine conjecture" in the spring of 1989. He had spent eight years thinking about this conjecture while producing many other highly significant results. He had the inspiration necessary to prove the conjecture while travelling in India [8]:-

I was in Kashmir. I had been travelling around northern India, and there was one really long tiresome bus journey, which lasted about 24 hours. Then the bus had to stop because there was a landslide and we couldn't go any further. It was all pretty darn unpleasant. Anyway, I was just toying with some calculations on this bus journey and finally I found an idea which made everything work.

The ideas behind the moonshine conjecture are hard to explain. Here is what Allyn Jackson writes in [5]:-

In the classification of finite simple groups, one of the most mysterious objects found was the monster group. There are various conjectures that attempt to connect the monster to other parts of mathematics. Borcherds invented the notion of a vertex algebra and used it to solve the Conway-Norton conjecture, which concerns the representation theory of the monster group (this theory is sometimes called "monstrous moonshine"). He used these results to generate product formulae for certain modular and automorphic forms. The first such formulae were found in the one-dimensional case by Euler and Jacobi, and the conventional wisdom in algebraic geometry was that such product formulae could not exist in higher dimensions. Borcherds's work is also important in physics, as it lays rigorous groundwork for conformal field theory in two dimensions.

Borcherds remained as a Royal Society University Research Fellow at Cambridge until 1992, and then spent the following year as a lecturer at Cambridge. In 1993 he was appointed as Professor of Mathematics at the University of California, Berkeley. He returned to Cambridge in 1996 and spent three years there as a Royal Society Professor in the Department of Mathematics before returning to his professorship at the University of California, Berkeley, in 1999. He continues to hold this position.

In 1992 Borcherds was awarded a Junior Whitehead Prize by the London Mathematical Society. The citation reads:-

A Junior Whitehead Prize 1992 is awarded to Dr Richard Borcherds of Cambridge University for his work on mathematical aspects of conformal field theory. Borcherds has made a number of important and original contributions to mathematics, with striking and surprising applications. One notable piece of work is his generalisation of the Kac-Moody algebras, to which much of classical representation theory applies. For one of the algebras in his class, the analogue of the Weyl Denominator Formula gives a remarkable, and quite unexpected, factorisation of the classical modular function as an infinite product. Another important concept invented by Borcherds is that of a 'vertex operator algebra'. This is a way of formulating the essential mathematical ideas of conformal quantum field theory. The concept was used extensively by Frenkel, Lepowski and Meurman in their work on the Monster simple group, and very recently it has been an essential tool in the progress made by E Frenkel on a conjecture of Drinfeld about the Langlands correspondence for representations of loop groups. In a marvellous synthesis of the ideas which he has been developing, Borcherds has proved conjectures of Conway and Norton on the 'Moonshine' module for the Monster group.

Also in 1992 he received a prize from the European Mathematical Society at the European Congress of Mathematicians in Paris. On 10 March 1994 he was elected to a fellowship of the Royal Society. His greatest honour, however, was being awarded a Fields Medal on 18 August 1998 at the Opening Ceremonies of the International Congress of Mathematicians in Berlin. The award was made:-

... for his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie Algebras, the proof of the Conway-Norton moonshine conjecture and the discovery of a new class of automorphic infinite products.

However, Borcherds took the fame which went with this honour in his stride [8]:-

Before the award I used to think it was terribly important, but now I realise that it's meaningless. However, I was over the moon when I proved the moonshine conjecture. If I get a good result I spend several days feeling really happy about it. I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine.

Borcherds is married to the topologist Ursula Gritsch and has two daughters. His current research interests include a mathematically rigorous approach to quantum field theory.