戴维·芒福德David Mumford

戴维·芒福德的父亲William Mumford是英国人，[2]：-

……一位具有国际视野的远见者，他在坦桑尼亚创办了一所实验学校，其理念基于适用技术……

芒福德的父亲从1945年联合国成立时起就在那里工作，这也是芒福德成长过程中他的职业。芒福德的母亲是美国人，一家人住在美国长岛海湾，这是北大西洋一个半封闭的海湾，北岸是纽约州和康涅狄格州海岸，南面是长岛。

在就读埃克塞特学校之后，芒福德进入哈佛大学。正是在哈佛，芒福德第一次对代数簇产生了兴趣。他在[2]中讲述道：-

……一位同学说“跟我去听奥斯卡·扎里斯基的第一堂课吧，哪怕我们一个字也听不懂”，而奥斯卡·扎里斯基把我迷住了。当他说出“代数簇”这几个字时，他的声音中有一种特别的共鸣，清楚地表明他正在窥视一座秘密花园。我立刻也想能够做到这一点。这让我花了25年苦苦挣扎，力图使这个世界变得可触可感、清晰可见。

从哈佛毕业后，芒福德被聘为该校教职员。他于1967年被任命为数学教授，十年后成为希金斯教授。他于1981年至1984年担任哈佛大学数学系主任，并于1987年至1992年担任麦克阿瑟会士。

芒福德最大的荣誉是1974年在温哥华国际数学家大会上被授予菲尔兹奖。约翰·泰特在[6]中描述了芒福德因之被授予约翰·查尔斯·菲尔兹奖章的工作。他写道：-

芒福德的主要工作是对模簇（即其点参数化某类几何对象的同构类的簇）的存在性和结构问题进行了极为成功的多管齐下的攻关。除此之外，他还对代数曲面理论做出了若干重要贡献。……芒福德在奥斯卡·扎里斯基之后，推进了将意大利学派关于代数曲面的工作代数化和严格化的计划。他为将费代里戈·恩里克斯的分类理论推广到特征p > 0的情形做了大量工作，在那里出现了许多新的困难。

约翰·泰特接着用更专业的术语解释了芒福德在[6]中的工作，这在[4]中也有描述。然而在20世纪80年代，芒福德的工作方向发生了戏剧性的变化。他在[1]中写道，在他的第一任妻子Erika去世后：-

……我从代数几何转向了一个旧日的爱好——是否存在一种数学方法来理解思维和大脑？这是应用数学，我必须说我认为定理在这里并不十分重要。我遇到了几位杰出的人，他们向我展示了统计学所起的关键作用：Grenander、Geman和佩尔西·戴康尼斯。

文章[3]是对芒福德所研究的这一新领域的综述。它读起来引人入胜，我们在此引用引言中的一些评论，这些评论可以让人了解所涵盖思想的范围：-

“模式理论”这一术语由Ulf Grenander在70年代引入，作为一个应用数学领域的名称，它为来自计算机视觉、语音识别、图像与声学信号处理、模式识别及其统计方面、神经网络以及人工智能部分领域的大量相关思想、技术和结果提供了理论框架。……“模式理论”旨在解决的问题……可以描述如下：“分析世界以任何模块性生成的模式，连同其所有自然出现的复杂性和歧义性，目标是重建产生这些模式的过程、对象和事件，并在这些模式再次出现时预测它们”。

芒福德除了获得菲尔兹奖章外，还获得了许多荣誉。他于1983年获得华威大学荣誉理学博士学位，2000年获得挪威科技大学荣誉理学博士学位，2001年获得洛克菲勒大学荣誉理学博士学位。他于1975年当选国家科学院会士，1978年当选塔塔基础研究所荣誉研究员，1991年当选罗马Accademia Nazionale dei Lincei外籍院士，1995年当选伦敦数学会荣誉院士，1997年当选美国哲学会会士，2008年当选皇家学会外籍院士。他于2006年获得邵逸夫奖，2007年获得美国数学会颁发的凯瑟琳·斯蒂尔数学阐述奖，2008年获得沃尔夫奖。他于1995年当选国际数学联盟主席，任职至1999年。

让我们提一下他与Caroline Series和弗洛伦斯·南丁格尔·大卫 Wright于2002年出版的Indra's Pearls: The Vision of Felix Klein一书。Vasily Chernecky对这本非凡著作的书评开头如下：-

菲利克斯·克莱因，十九世纪伟大的几何学家之一，在数学中发现了一个在佛教神话中已有预示的思想：因陀罗的天堂包含一张珍珠网，每颗珍珠都映现在其邻珠之中，因此整个宇宙都映照在每一颗珍珠里。菲利克斯·克莱因研究了无限重复的反射，并由此引出了具有多重共存对称性的形式，每一种形式本身都很简单，但它们的相互作用在混沌边缘产生了分形。一个世纪以来，这些几乎不可能手工绘制的图像，几乎只存在于数学家的想象之中。然而在20世纪80年代，作者们开始了对菲利克斯·克莱因愿景的首次计算机探索，并在此过程中发现了他们自己的更多非凡图像。本书写作的目的是作为实际编写用于生成精美分形花边图案的算法的指南，其中大部分图案此前从未印刷出版过。

最后，为了了解他当前的兴趣，我们提一下他于2008年2月11日发表的一场演讲。演讲题为What's an infinite dimensional manifold and how can it be useful in hospitals?，芒福德给出了以下摘要：-

变形人脸已成为一种流行的游戏，但其背后的数学是什么？一种看待它的方式是在无限维形状流形上构造测地线。我将尝试用简单的例子来解释这意味着什么，然后说明为什么它被提议作为诊断医疗状况的新工具。

David Mumford's father, William Mumford, was English and [2]:-

... a visionary with an international perspective, who started an experimental school in Tanzania based on the idea of appropriate technology...

Mumford's father worked for the United Nations from its foundations in 1945 and this was his job while Mumford was growing up. Mumford's mother was American and the family lived on Long Island Sound in the United States, a semi-enclosed arm of the North Atlantic Ocean with the New York- Connecticut shore on the north and Long Island to the south.

After attending Exeter School, Mumford entered Harvard University. It was at Harvard that Mumford first became interested in algebraic varieties. He relates in [2]:-

... a classmate said "Come with me to hear Professor Zariski's first lecture, even though we won't understand a word" and Oscar Zariski bewitched me. When he spoke the words "algebraic variety", there was a certain resonance in his voice that said distinctly that he was looking into a secret garden. I immediately wanted to be able to do this too. It led me to 25 years of struggling to make this world tangible and visible.

After graduating from Harvard, Mumford was appointed to the staff there. He was appointed professor of mathematics in 1967 and, ten years later, he became Higgins Professor. He was chairman of the Mathematics Department at Harvard from 1981 to 1984 and MacArthur Fellow from 1987 to 1992.

Mumford's greatest honour was being awarded a Fields Medal at the International Congress in Vancouver in 1974. Tate describes the work that Mumford was awarded the Fields Medal for in [6]. He writes:-

Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parameterise isomorphism classes of some type of geometric object. Besides this he has made several important contributions to the theory of algebraic surfaces. ... Mumford has carried forward, after Zariski, the project of making algebraic and rigorous the work of the Italian school on algebraic surfaces. He has done much to extend Enriques' theory of classification to characteristic p > 0, where many new difficulties appear.

Tate goes on to explain in more technical terms Mumford's work in [6] and it is also described in [4]. In the 1980s however, the direction of Mumford's work changed dramatically. He writes in [1] that, after his first wife Erika died:-

... I turned from algebraic geometry to an old love - is there a mathematical approach to understanding thought and the brain? This is applied mathematics and I have to say that I don't think theorems are very important here. I met remarkable people who showed me the crucial role played by statistics, Grenander, Geman and Diaconis.

The article [3] is a survey of this new area that Mumford has worked on. It makes fascinating reading and we quote here some comments from the introduction which give an idea of the scope of the ideas covered:-

The term "Pattern Theory" was introduced by Ulf Grenander in the 70's as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence. ... The problem that "Pattern Theory" aims to solve ... may be described as follows "the analysis of the patterns generated by the world in any modularity, with all their naturally occurring complexity and ambiguity, with the goal of reconstructing the processes, objects and events that produced them and of predicting these patterns when they reoccur".

Mumford has received many honours in addition to the Fields Medal. He received an honorary D.Sc. from the University of Warwick in 1983, an honorary D.Sc. from the Norwegian University of Science and Technology in 2000, and an honorary D.Sc. from Rockefeller University in 2001. He was elected to the National Academy of Sciences in 1975, elected an Honorary Fellow of the Tata Institute of Fundamental Research in 1978, elected a Foreign Member of Accademia Nazionale dei Lincei, Rome, in 1991, elected an Honorary Member of London Mathematical Society in 1995, elected to the American Philosophical Society in 1997, and elected a Foreign Member of the Royal Society in 2008. He was awarded the Shaw Prize in 2006, the Steele Prize for Mathematical Exposition by the American Mathematical Society in 2007, and the Wolf Prize in 2008. He was elected President of the International Mathematical Union in 1995, a position he held until 1999.

Let us mention the book Indra's Pearls: The Vision of Felix Klein which he published with Caroline Series and David Wright in 2002. Vasily Chernecky begins a review of this remarkable book as follows:-

Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeated reflections and was led to forms with multiple co-existing symmetries, each simple in itself, but whose interactions produce fractals on the edge of chaos. For a century these images, which were practically impossible to draw by hand, barely existed outside the imagination of mathematicians. However in the 1980s the authors embarked on the first computer exploration of Klein's vision, and in so doing found further extraordinary images of their own. The book is written as a guide to actually coding the algorithms which are used to generate the delicate fractal filigrees, most of which have never appeared in print before.

Finally, to get a feeling for his current interests we mention a lecture he delivered on 11 February 2008. The lecture was entitled What's an infinite dimensional manifold and how can it be useful in hospitals? and Mumford gave the following abstract:-

Morphing faces has become a popular game but what is the math behind it? One way to view it is as the construction of geodesics on an infinite dimensional manifold of shapes. I will try to explain what this means, using simple examples and then go on show why it is proposed as a new tool in the diagnosis of medical conditions.