丹尼尔·奎伦Daniel Quillen

丹尼尔·奎伦的父亲，Charles Sylvester 奎伦（1909-1989），接受过化学工程师培训，但以高中物理教师的职业为生。他的母亲，Emma Lewis Gray（1910-1999），是一名秘书。丹，正如他广为人知的那样，是他父母两个儿子中的长子。Jean Quillen在[7]中写道：-

从小，他的智力才能和对待世界的独特方式就显而易见。他的母亲过去喜欢讲丹在婴儿时期不说话，直到他能说出完整的句子，让所有人都大吃一惊。……他的母亲在他幼年生活中是一股推动力，促使他人认识到他的才能。

在他十二岁时父亲给他一本微积分书之后，他才对数学产生了热情。有一段时间他也热衷于国际象棋，但当他觉得竞争性下棋过于紧张时，他对下棋的兴趣就减弱了。他获得了一笔奖学金，使他能够进入纽瓦克学院，一所评价很高的私立中学，并且在他本应在纽瓦克学院完成学业的前一年，他获得了在哈佛大学学习数学的奖学金。丹的母亲发现哈佛有一个特殊项目，让有才华的学生比正常情况提前一年开始学习，所以，赢得了奖学金后，他能够跳过中学最后一年，于1957年9月进入哈佛。尽管他没有完成纽瓦克学院的文凭课程，然而，由于他的表现如此出色，他还是被授予了文凭。作为哈佛大学的学生，奎伦参加了1959年第20届个人和团体普特南竞赛，并被命名为普特南会士。他的女友琼（他后来娶了她）写道[7]：-

……他似乎像海绵一样吸收他的本科数学课程。他学习并理解了所修每一门课程的全部内容。事实上，我注意到他几乎能凭记忆复现每一个定理和证明。他还有能力识别一门学科中重要的内容。有一次当我在一门课程上落后时，他设法在仅仅三天内教会了我所有要点。我不仅在考试中得了A，还注意到试卷上有一处印刷错误！

他于1961年获得文学学士（最优等成绩），并在次年获得硕士学位。然后他开始在哈佛大学在拉乌尔·博特的指导下进行研究。奎伦觉得拉乌尔·博特在他作为数学家的成长中发挥了巨大作用。Graeme 尔文·席格写道（见[7]或[14]）：-

他说拉乌尔·博特——一个身材高大、性格外向的人，因其热情和个人魅力而广受爱戴，外表上与他害羞而沉默的学生截然相反——对他来说是一个至关重要的榜样，向他表明一个人不必反应敏捷才能成为杰出的数学家。与拉乌尔·博特不同，后者表演式地让人把一切解释给他听许多遍，奎伦在别人看来一点也不慢，然而他视自己为一个必须从第一原理出发非常缓慢而仔细地思考事情，并且必须为每一点进步而努力工作的人。他对自己的能力 truly 谦虚——非常迷人地谦虚——尽管同时又有雄心和动力。

迈克尔·阿蒂亚第一次见到奎伦是在后者在哈佛大学攻读博士学位时。他写道[7]：-

我记得一个容易激动的年轻人，满脑子想法和热情，拉乌尔乐于鼓励他。

1961年6月3日，奎伦还在哈佛大学攻读硕士学位、准备攻读博士学位时，与Jean Plesset结婚。他们第一次见面是在奎伦本科第二年选修一年级化学课时，但正是他们对音乐的共同热爱使他们走到了一起，因为他们通过哈佛管弦乐团定期见面。Jean演奏中提琴，奎伦总是声称自己演奏三角铁。然而，Jean总是声称奎伦是乐团的图书管理员，偶尔吹小号。席格写道（见[7]或[14]）：-

他喜欢“弄清楚”音乐如何运作，也喜欢创作二三十小节的微型作品，但他在数学上太有驱动力，不允许自己花太多时间在音乐上。他和Jean在他完成博士学位前有了两个孩子，后来总共有了六个孩子。

事实上，Jean打写了她丈夫的博士论文。她写道[7]：-

我记得打写他的博士论文：Dan在一个房间产出页面，我在隔壁房间用租来的电动打字机。我们怎么应付两个婴儿，我无法想象！自然，论文是在到期后大约三分钟才完成的。

1964年，奎伦因其关于偏微分方程的学位论文Formal Properties of Over-Determined Systems of Linear Partial Differential Equations而获得博士学位。获得博士学位后，奎伦被任命为麻省理工学院教员。他在其他大学进行了多年研究，这些研究对确定他的研究方向至关重要。1968-69学年，他是巴黎高等科学研究所的斯隆研究员，当时深受亚历山大·格罗滕迪克影响。Hyman Bass写道[7]：-

在巴黎的那一年里，奎伦展现了他典型的个人特质：温和善良的天性、谦逊、随性而孩子气的外表，并未因早生的白发而改变，以及他已然充实的家庭生活。在那个才华横溢且常常光彩夺目的数学圈子里，奎伦似乎听得多、说得少，而且只在有实质内容可说时才开口。他后来的工作表明他是一个深刻的倾听者。

1969-70年间，奎伦是普林斯顿高等爱德华·斯图迪研究所的访问成员，当时他深受迈克尔·阿蒂亚的影响；1973-74年间，他再次作为古根海姆研究员访问巴黎高等科学研究所。Andrew Ranicki也在同一年在高等科学研究所度过，他写道[7]：-

Dan和他的妻子Jean都对我很好，我经常造访Ormaille住宅区的8号馆。虽然我和Dan谈论数学并不多，但有很多其他话题，我总是对他目标的严肃性和思想的独立性印象深刻，再加上他迷人的个人谦逊。

丹尼斯·苏利文也在1973-74年间在巴黎高等科学研究所度过。他写道[7]：-

有一个记忆似乎与一切都吻合：在Ormaille住宅区8号馆主房间中央，有一张没有椅子的大而光滑的木桌。桌上数百个小形状被部署成十几个整齐的小方阵，围绕着中间正在形成的一个连贯结构。Dan和几个孩子以及任何可能在周围的人都在桌子周围徘徊，专注地凝视这些图案，轻声低语，希望体验到将新部分融入正在形成的结构中的美妙乐趣。这是一件严肃的事情，有着良好的氛围。

在20世纪60年代，奎伦描述了如何定义单纯对象的同调，涵盖许多不同的范畴，包括集合、环上的代数，以及诺曼·斯廷罗德代数上的不稳定代数。弗兰克·亚当斯在同伦论中提出了一个猜想，奎伦对此进行了研究。奎伦用两种截然不同的方法处理弗兰克·亚当斯猜想，即使用来自代数几何的技巧，以及使用群的模表示论的技巧。两种方法都被证明是成功的，第一种方法的证明由奎伦的一名学生完成，第二种方法导致奎伦本人给出了证明。使用群的模表示论的技巧被奎伦在后来关于群的上同调和代数K理论的工作中大量使用。上同调的工作导致奎伦给出了有限群的mod p上同调环的结构定理，这个结构定理解决了该领域中的许多未决问题。

奎伦于1978年在赫尔辛基举行的国际数学家大会上获得了约翰·查尔斯·菲尔兹奖章。他因在1972年作为高阶代数$K$-理论的主要奠基人而获奖，这一新工具成功地运用几何与拓扑的方法和思想来表述并解决代数中的重大问题，特别是环论与模论中的问题。代数K-理论是亚历山大·格罗滕迪克的思想在交换环上的推广。亚历山大·格罗滕迪克的思想被迈克尔·阿蒂亚和弗里德里希·希策布鲁赫用来创建拓扑$K$-理论。显然，奎伦在巴黎受亚历山大·格罗滕迪克影响的那一年以及在普林斯顿与迈克尔·阿蒂亚共事的那段时间，是奎伦发展代数K-理论的重要因素。

Bass在[2]中描述了奎伦如何解决这样一个问题：高阶代数$K$-群，即$n ≥ 3$的$K_{n}$，其构造方式与亚历山大·格罗滕迪克构造本质上不同，这带来了巨大困难：-

……他从同伦论中借用了技巧，而且是以一种完全新颖的方式。这篇出现所谓Q-构造的论文，本质上没有数学上的先驱。初次阅读它，就像降落在一个崭新而友好的数学星球上。在那里遇到的不仅是新定理和新方法，还有新的数学生物以及一整套处理它们的姿态范式。高阶代数K-理论实际上在那里从基本原理出发被建立起来，并且在63页内就达到了人们通常期望多位数学家经过数年努力才能达到的成熟状态。

当奎伦得知自己将获得菲尔兹奖时，Jeanne Duflot正在以他为导师进行她的博士学位研究。她写道[7]：-

他是个滴酒不沾的人，而在我还是他学生的时候，他获得了约翰·查尔斯·菲尔兹奖章；我得知这件事后，在第一次见面时敬畏得几乎说不出话，而当他送给我一瓶香槟时，我更是惊呆了——那是一位祝贺他的同事送给他的，他解释说他不喝含酒精的饮料。我感激地接受了它，并和一些同学一起喝了。我觉得那是相当不错的香槟，但我并不是行家。

至于他的性格，这在[2]中有所体现：-

奎伦在24岁时获得了博士学位，他和妻子Jean——一位小提琴家——当时已经在照料他们五个孩子中的两个[他们现在有六个孩子]。他作为数学家和作为父亲的早熟，也许促成了他头发过早变白，但这并没有改变他孩子气的相貌，也没有改变他随和而谦逊的举止。他过着一种有些退隐的生活方式，很少公开露面，而一旦露面，几乎总是带着某个非凡的新定理或新想法。

在2中，Hyman Bass对奎伦在1978年获得约翰·查尔斯·菲尔兹奖章之前的贡献总结如下：-

数学才能往往表现为解决问题或构建理论。像奎伦这样的罕见情况，人们才能满意地看到，用具有强大力量和广泛范围的普遍思想，以及通过统一来自数学不同领域的方法，解决了困难而具体的问题。奎伦对整整一代年轻代数学家和拓扑学家的观念和思维习惯产生了深远影响。人们研究他的工作不仅是为了获取信息，更是为了受到启迪。

从1984年到2006年，奎伦是牛津大学莫德林学院的韦恩弗利特纯数学讲席教授。Jean Quillen解释了他去牛津的想法是如何开始的，当时他正在波恩的马克斯·普朗克研究所进行为期一年的学术休假[7]：-

正是丹与迈克尔·阿蒂亚的关系首先把我们带到了牛津。在波恩的那一年，丹对我说：“我待错了国家。”我说：“你应该在哪个国家？”他说他想在牛津，部分原因是他对迈克尔·阿蒂亚正在研究的东西感兴趣。在牛津待了一年后，我们回到了波士顿。大约六个月后，丹在牛津做报告时，迈克尔·阿蒂亚提到莫德林学院的韦恩弗利特讲席即将空缺，并问他是否考虑永久搬到牛津。他打电话给我。我说：“是的，请。”这就是我们来到牛津并留下来的经过。

2000年，期刊K-Theory发行了一期特刊，专门献给奎伦，以庆祝他的六十岁生日。同年，他在佛罗里达大学就Module theory for nonunital rings做了埃尔德什学术报告。2006年5月22日，牛津举办了“第39届K理论日”，以庆祝奎伦的65岁生日。Jacek Brodzki就Analysis and geometry on discrete groups做了报告，Mathai Varghese就T-duality and non-commutative geometry做了报告，Joachim Cuntz就K-theory for locally convex algebras做了报告，Eric Friedlander以报告Dan and me: looking back at some of Dan's remarkable mathematics结束了会议。2006年，奎伦在年满65岁后退休。

奎伦去世之前，经历了极其悲伤的几年，在此期间阿尔茨海默病对他造成了越来越大的损害。Jean Quillen写道[12]：-

阿尔茨海默病确实是一种可怕的疾病。在过去5年多的时间里，看着它如何影响丹，实在令人难以承受。它夺走了他所热爱的事物，也过早地将他从我身边带走。它首先夺走了他从事数学的能力，接着是演奏音乐、阅读、理性思考的能力，最后是辨认他所爱之人的能力（谢天谢地，除了我）。

对于一位真正伟大的数学家来说，这是多么令人悲伤的结局。2010年6月21日The Gainsville Sun中的一篇报道说明了其中的困难，报道称奎伦失踪了[5]：-

盖恩斯维尔警方周一傍晚正在搜寻一名失踪男子，该男子是阿尔茨海默病患者。69岁的奎伦 Gray Quillen可能正前往韦斯特赛德公园……警员们在该地区寻找奎伦，据描述他身高5英尺6英寸，体重160磅，头发灰白稀疏。他戴眼镜。警方提供了衣着描述，称奎伦应穿着绿色格子衬衫、卡其色短裤和凉鞋。

他在生命的最后一周于黑文临终关怀医院接受照护，并因阿尔茨海默病晚期相关并发症在那里去世。

在授予奎伦的众多荣誉中，除了约翰·查尔斯·菲尔兹奖章外，我们还提到，他曾于1974年8月在温哥华举行的国际数学家大会上作全会报告，当时他作了题为“高等代数$K$-理论”的演讲。1975年，American Mathematical Society因以下工作授予他法兰克·尼尔森·寇尔代数奖：-

……因其论文《高等代数K-理论》。

1983年，他应邀在阿伯丁举行的英国数学学术讨论会上作全会报告，当时他作了Infinite determinants over algebraic curves arising from problems in geometry, differential equations and number theory讲座。

让我们以Journal of K-Theory[4]的编辑对奎伦的悼念来结束这篇传记：-

他比任何人都更有功于创建了今日所研究的代数K-理论这一学科，并展示了它的力量与优雅。他还在数学的许多其他方面做出了根本性的贡献：有理同伦、模型范畴、形式群和循环同调，仅举几例。他所发展的所有思想都将在他身后留存，并赋予他20世纪伟大数学家的地位。许多数学家，包括我们编辑委员会的所有成员，都深受他的远见、教学和写作的启发和影响。作为致力于奎伦 largely created这一学科的编辑，我们高度赞赏他对期刊“K-Theory”及其后继“Journal of K-Theory”的关键支持，以及他为我们的数学领域所做的一切。我们将深深怀念他，并深情地铭记他。

Daniel Quillen's father, Charles Sylvester Quillen (1909-1989), trained as a chemical engineer but made his career as a high school physics teacher. His mother, Emma Lewis Gray (1910-1999), was a secretary. Dan, as he was widely known, was the eldest of his parents' two sons. Jean Quillen writes in [7]:-

From an early age his intellectual abilities and particular approach to the world were apparent. His mother used to enjoy telling how Dan didn't talk as a baby until he surprised everyone by being able to speak full sentences. ... His mother was a driving force in his young life, pushing others to recognize his abilities.

It was after his father gave him a calculus book when he was twelve years old, that he developed a passion for mathematics. For a while he was also passionate about chess but his interest in that diminished when he felt that playing competitively was too intense. He won a scholarship which enabled him to attend Newark Academy, a highly rated private secondary school, and, a year before he should have completed his schooling at Newark Academy, he won a scholarship to study mathematics at Harvard University. Dan's mother had discovered that Harvard had a special programme for talented students to begin their studies a year earlier than normal so, having won the scholarship, he was able to miss out the last year of secondary school and enter Harvard in September 1957. Although he hadn't completed the work for his diploma at Newark Academy, nevertheless, because he performance was so outstanding, he was awarded his diploma. As a Harvard University student, Quillen took part in the 20th Individual and Team Putnam Competition in 1959 and was named as a Putnam Fellow. His girlfriend Jean (whom he later married) wrote [7]:-

... he seemed to absorb his undergraduate mathematics courses like a sponge. He learned and understood all of every course he took. In fact I noticed that he could reproduce by memory nearly every theorem and proof. He also had the talent of being able to identify what was important in a subject. Once when I got behind in a course, he managed to teach me all the important points in only three days. Not only did I get an A on the exam, but I also noticed a misprint in the examination paper!

He received his B.A. (magna cum laude) in 1961 and received his Master's degree in the following year. He then began research at Harvard under Raoul Bott's supervision. Quillen felt that Bott had played a huge part in his development as a mathematician. Graeme Segal writes (see [7] or [14]):-

He said that Bott - a large, outgoing man universally beloved for his warmth and personal magnetism, outwardly quite the opposite of his shy and reticent student - was a crucial model for him, showing him that one did not have to be quick to be an outstanding mathematician. Unlike Bott, who made a performance of having everything explained to him many times over, Quillen did not seem at all slow to others, yet he saw himself as someone who had to think things out very slowly and carefully from first principles and had to work hard for every scrap of progress he made. He was truly modest about his abilities - very charmingly so - though at the same time ambitious and driven.

Michael Atiyah first met Quillen when he was working for his Ph.D. at Harvard. He writes [7]:-

I remember an excitable young man bubbling with ideas and enthusiasm which Raoul was happy to encourage.

While just beginning to study for his Master's Degree at Harvard for his Ph.D., on 3 June 1961, Quillen married Jean Plesset. They first saw each other when Quillen took a first year chemistry class in his second year of undergraduate study, but it was their common love of music that brought them together for they met regularly through the Harvard orchestra. Jean played the viola and Quillen always claimed that he played the triangle. Jean, however, always claimed that Quillen was the orchestra's librarian and occasional trumpeter. Segal writes (see [7] or [14]):-

He delighted in "figuring out" things about how music worked and in devising tiny compositions of twenty or thirty bars, but he was far too driven mathematically to let himself spend much time on music. He and Jean had two children before he completed his Ph.D. and went on to have six altogether.

In fact Jean typed her husband's Ph.D. thesis. She wrote [7]:-

I remember typing his Ph.D. thesis: Dan in one room producing pages, me in the next room with a hired electric typewriter. What we did with the two babies I cannot imagine! Naturally the thesis was produced about three minutes after it was due.

In 1964, Quillen was awarded his Ph.D. for his thesis on partial differential equations entitled Formal Properties of Over-Determined Systems of Linear Partial Differential Equations. After receiving his doctorate, Quillen was appointed to the faculty of Massachusetts Institute of Technology. He spent a number of years undertaking research at other universities which were to prove important in setting the direction of his research. He was a Sloan Fellow at the Institut des Hautes Études Scientifiques in Paris during academic year 1968-69 when he was greatly influenced by Alexander Grothendieck. Hyman Bass writes [7]:-

During the year in Paris, Quillen presented his typical personal characteristics: a gentle good nature, modesty, a casual and boyish appearance unaltered by his prematurely graying hair, and his already ample family life. In that brilliant, and often flamboyant, mathematical milieu, Quillen seemed to listen more than he spoke, and he spoke only when he had something substantial to say. His later work showed him to be a deep listener.

Quillen was a visiting member of the Institute for Advanced Study at Princeton during 1969-70 when he was strongly influenced by Michael Atiyah, and a Guggenheim Fellow, again at the Institut des Hautes Études Scientifiques in Paris, during 1973-74. Andrew Ranicki also spent the same year at the Institut des Hautes Études Scientifiques and writes [7]:-

Both Dan and his wife, Jean, were kind to me, and I was a frequent visitor at Pavillon 8 of the Residence de l'Ormaille. Although I did not talk to Dan all that much about mathematics, there were plenty of other topics, and I was always impressed by his seriousness of purpose and independence of mind, allied with a winning personal modesty.

Dennis Sullivan also spent the year 1973-74 at the Institut des Hautes Études Scientifiques in Paris. He writes [7]:-

One memory that seems to fit with everything was of a large smooth wooden table situated without chairs in the middle of the main room of Pavillon 8 in the Residence d'Ormaille. On the table hundreds of little shapes were deployed into a dozen or so neat little battalions surrounding a coherent structure emerging in the middle. Dan and a couple of kids and anyone else who might be around were hovering around the table, peering intently at these patterns, muttering softly and hoping to experience the exquisite pleasure of fitting in new parts to the emerging structure. It was serious business with good karma.

In the 1960s, Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra. Frank Adams had formulated a conjecture in homotopy theory which Quillen worked on. Quillen approached the Adams conjecture with two quite distinct approaches, namely using techniques from algebraic geometry and also using techniques from the modular representation theory of groups. Both approaches proved successful, the proof in the first approach being completed by one of Quillen's students, the second approach leading to a proof by Quillen himself. The techniques using modular representation theory of groups were used by Quillen to great effect in later work on cohomology of groups and algebraic K-theory. The work on cohomology led to Quillen giving a structure theorem for mod p cohomology rings of finite groups, this structure theorem solving a number of open questions in the area.

Quillen received a Fields Medal at the International Congress of Mathematicians held in Helsinki in 1978. He received the award as the principal architect of the higher algebraic $K$-theory in 1972, a new tool that successfully used geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory. Algebraic K-theory is an extension of ideas of Grothendieck to commutative rings. Grothendieck's ideas were used by Atiyah and Hirzebruch when they created topological $K$-theory. Clearly Quillen's year spent in Paris under Grothendieck's influence and at Princeton working with Atiyah were important factors in Quillen's development of algebraic K-theory.

Bass describes in [2] how Quillen resolved the problem that the higher algebraic $K$-groups, $K_{n}$ for $n ≥ 3$, being constructed in an essentially different way from the Grothendieck construction presented great difficulties:-

... he borrowed techniques from homotopy theory, and in a completely novel way. The paper in which this so-called Q-construction occurs is essentially without mathematical precursors. Reading it for the first time is like landing on a new and friendly mathematical planet. One meets there not only new theorems and new methods, but new mathematical creatures and a complete paradigm of gestures for dealing with them. Higher algebraic K-theory is effectively built there from first principles and, in 63 pages, reaches a state of maturity that one normally expects from the efforts of several mathematicians over several years.

Jeanne Duflot was undertaking research for her Ph.D. with Quillen as her advisor when he learnt that he would receive the Fields medal. She writes [7]:-

He was a teetotaler, and he won the Fields Medal while I was his student; I was awestruck and could barely speak to him at our first meeting after I found out about this and was floored when he offered me a bottle of champagne that had been given to him by a congratulatory colleague, explaining that he did not drink alcoholic beverages. I gratefully accepted it and drank it with some fellow students. I think it was quite good champagne, but I was not an expert.

As to his character, this is shown in [2]:-

When Quillen received his Ph.D. at the age of 24, he and his wife Jean, a violinist, were already caring for two of their five children [they now have six]. His precocity as a mathematician and as a father perhaps influenced the early greying of his hair, but it has not altered his boyish look or his easy and modest manner. He has a somewhat retiring life-style, appearing rarely in public, and then almost invariably with some extraordinary new theorem or idea in hand.

In [2] Hyman Bass sums up Quillen's contribution leading up to the award of the Fields Medal in 1978 as follows:-

Mathematical talent tends to express itself either in problem solving or in theory building. It is with rare cases like Quillen that one has the satisfaction of seeing hard, concrete problems solved with general ideas of great force and scope and by the unification of methods from diverse fields of mathematics. Quillen has had a deep impact on the perceptions and the very thinking habits of a whole generation of young algebraists and topologists. One studies his work not only to be informed, but to be edified.

From 1984 to 2006 Quillen was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Jean Quillen explained how his idea to go to Oxford began when he was spending a year study leave at the Max Planck Institute in Bonn [7]:-

It was Dan's relationship with Atiyah that first brought us to Oxford. During the year in Bonn, Dan said to me, "I'm in the wrong country." I said, "What country should you be in?" He said he wanted to be in Oxford, partly because he was interested in something that Michael Atiyah was working on. After a year in Oxford we returned to Boston. About six months later Dan was in Oxford giving a talk when Atiyah mentioned that the Waynflete Chair at Magdalen College was opening up and asked if he would consider moving to Oxford permanently. He phoned me. I said, "Yes, please," and that's how we came to Oxford to stay.

In 2000 the journal K-Theory issued a special part dedicated to Quillen on the occasion of his sixtieth birthday. In the same year he gave the Erdős Colloquium at the University of Florida on Module theory for nonunital rings. On 22 May 2006 the '39th K-theory Day' at Oxford was set up to celebrate Quillen's 65th birthday. Jacek Brodzki lectured on Analysis and geometry on discrete groups, Mathai Varghese lectured on T-duality and non-commutative geometry, Joachim Cuntz lectured on K-theory for locally convex algebras, and Eric Friedlander closed the proceedings with the lecture Dan and me: looking back at some of Dan's remarkable mathematics. Having reached the age of 65, Quillen retired in 2006.

Quillen's death came after an extremely sad number of years during which Alzheimer's disease took an increasing toll on him. Jean Quillen writes [12]:-

Alzheimer's is truly a terrible disease. It was very hard to watch what it did to Dan over the past 5+ years. It stole the things he loved and took him from me too early. It first took his ability to do mathematics, then ability to play music, read, rational thinking and finally recognition of those he loved (except me, thank goodness).

What a sad end for a truly great mathematician. An illustration of the difficulties is seen in a report in The Gainsville Sun of 21 June, 2010 stating that Quillen was missing [5]:-

Gainesville police were searching Monday evening for a missing man who is an Alzheimer's patient. Daniel Gray Quillen, 69, may have been heading toward Westside Park ... officers were in the area looking for Quillen, who is described as 5-foot-6, 160 pounds with thinning gray hair. He wears glasses. Police provided a clothing description that said Quillen should be wearing a green plaid shirt, khaki shorts and sandals.

He was cared for at the Haven Hospice during the last week of his life, where he died from the complications associated with the final stages of Alzheimer's.

Among the many honours given to Quillen, in addition to the Fields Medal, we mention that he was a plenary speaker at the International Congress of Mathematicians at Vancouver in August 1974 when he gave the lecture 'Higher Algebraic $K$-Theory'. He was awarded the Cole Prize in Algebra by the American Mathematical Society in 1975 for:-

... for his paper "Higher algebraic K-theories".

He was an invited plenary speaker at the British Mathematical Colloquium in Aberdeen in 1983 when he gave the lecture Infinite determinants over algebraic curves arising from problems in geometry, differential equations and number theory.

Let us end this biography by quoting the tribute to Quillen from the editors of the Journal of K-Theory [4]:-

More than anyone else, he was responsible for creating the subject of algebraic K-theory as it is pursued today, and for demonstrating its power and elegance. He also made fundamental contributions to many other aspects of mathematics: rational homotopy, model categories, formal groups, and cyclic homology, to mention a few. All of the ideas he has developed will survive him and give him the stature of a great mathematician of the 20th century. Many mathematicians including all of the members of our Board were greatly inspired and influenced by his vision, his teaching and his writing. As editors devoted to the subject that Quillen largely created, we are highly appreciative of his crucial support for the journal "K-Theory" and its successor the "Journal of K-Theory", and of all that he has done for our area of mathematics. He will be greatly missed and fondly remembered.