丹尼斯·苏利文Dennis Sullivan

丹尼斯·苏利文 实际上应该是 苏利文 III，因为他的父亲和祖父都叫 苏利文。他的父亲，被称为 苏利文 Jr（1920-1978），于1920年7月27日出生在密歇根州的休伦港。他在密歇根州韦恩县迪尔伯恩的六角工具与工程公司工作。苏利文 Jr于1940年3月25日在密歇根州圣克莱尔县休伦港与丽塔·简·特林德（1922-2014）结婚。丽塔出生于1922年6月16日，是百货商店经理查尔斯·G·特林德及其妻子弗朗西斯的女儿。苏利文和丽塔苏利文有两个孩子，苏利文（1941年2月12日出生，本传记的主人公）和迈克尔Francis Sullivan（1942年7月22日出生）。迈克尔在长期与病魔斗争后，于1957年7月26日在德克萨斯州休斯顿去世，年仅十五岁。丽塔苏利文于1945年9月11日提出离婚，并于1946年8月7日获得绝对离婚判决。

苏利文出生于密歇根州休伦港，由母亲在得克萨斯州休斯顿抚养长大，一直自认为是得克萨斯人。2014年他母亲去世时，她的讣告写道[48]：-

苏利文是休斯顿的长期居民，其职业生涯跨越广播、电视、广告和房地产，于2014年8月9日星期六上午去世。

他在休斯顿上学，之后进入同样位于休斯顿的莱斯大学学习化学。他在[58]中解释说，尽管高中老师建议他不要申请莱斯大学，他还是申请了：-

我在高中时算是个差生，因为我不学习。我想这是因为我其实近视。我不知道你应该能看见黑板。所以当我告诉老师我要申请莱斯大学时，那是一所位于我居住的得克萨斯州休斯顿的非常好的学校，她说：“你不应该申请那里。你不够优秀。”

当他发现他所修的数学课程是所有科学课程中最令人兴奋的时，他从化学转到了数学[58]：-

我们学了很多科学。我不知道还有数学家这样的人。然后，在我的二年级数学课上，教授[已故的]Guy Johnson向我们展示了一个奇妙的数学定理，它远远超出了方程和计算等事物的智力水平，而这是我以为数学就是的东西。我对此非常震撼。一旦你理解了更深层的真正数学，事物的意义就变得惊人地清晰。没有歧义，而几乎所有其他事物与数学相比都更加模糊和晦涩。这其中有很大的安慰。这就是吸引我的地方。

把苏利文变成数学家的教授是Guy Johnson Jr（1922-2017）。他于1955年从莱斯大学获得博士学位，其两部分学位论文为第一部分：Collective Singularities of Families of Analytic Functions。第二部分：Regions of Flatness for Analytic Functions and their Derivatives ，由索勒姆·曼德尔布罗伊特作为他的学位论文导师。由于这一事件如此关键，我们给出更多细节。Guy Johnson [26]：-

……画了一个肾形游泳池和一个圆形游泳池的图。他说，你可以把这个肾形游泳池变形为圆形游泳池。在每个点，变形都是通过缩放进行的。这一点上的一个小三角形映射到另一点上的一个相似三角形。在每个点上，这都是真的。我们有这个映射的公式，因为我们在学微积分，我们有讨论它的记号，我们一直在学习。但这就像一个几何图像。这个映射本质上是唯一的。而且，这个陈述的性质与我以前见过的任何数学陈述都完全不同。它是，就像，普遍的、深刻的，哇！而且是真的！所以，几周之内，我把我的专业转到了数学。

他在莱斯大学攻读数学专业时学到的一个教训，他觉得这对他的研究方法很重要[64]：-

……我们有一次作业，上面有一道题谁也做不出来，所以不知为什么我就一直做下去。那是一道非常奇怪的题，两周后我做出来了。但我并不是班上最好的学生，班上有好几个非常优秀的学生。就在那时我意识到，如果你真的想理解某件事，你就必须坚持不懈地钻研，最终你会理解它，基本上就是这样。所以之后你必须找到一些简单的、可以着手的东西，这样你就不会被复杂性和信息完全弄糊涂。

苏利文于1963年获得莱斯大学颁发的学士学位。他于1963年8月10日在得克萨斯州哈里斯与Kathleen Rose McGuire结婚。随后他前往普林斯顿大学攻读数学研究生。在普林斯顿，他的学位论文导师是威廉·布劳德，苏利文于1966年因其学位论文Triangulating Homotopy Equivalences获得博士学位。Anthony Phillips写道[44]：-

当时威廉·布劳德的学生们正在从割补理论中挖掘拓扑学的金矿。苏利文的学位论文就属于这一路数，并引出了他关于主猜想的工作（1967）。

正是这项工作使苏利文于1971年从美国数学会获得奥斯瓦尔德·维布伦几何学奖。该奖项的第六次颁发情况如下[61]：-

苏利文因其关于主猜想的工作而获奖，该工作总结于论文《论流形的主猜想》，《美国数学会通报》，第73卷（1967）……

获得博士学位后，苏利文被任命为英格兰华威大学的北约研究员。我[EFR]当时是华威大学的研究生，我记得许多个晚上，我和 fellow research students 一起打桥牌，而苏利文则在邻桌撰写数学论文。在华威大学担任研究员之后，苏利文在乔治·伯克利的加利福尼亚大学担任Miller研究员，研究弗兰克·亚当斯猜想、K理论和étale同伦。1969年，他前往麻省理工学院，担任斯隆数学研究员，在那里[40]：-

……他的工作集中于他命名为几何拓扑的领域（特别是对埃瓦里斯特·伽罗瓦对称性的研究）以及利用微分形式为流形的有理同伦型构造极小模型。

1970年，他撰写了一套题为Geometric topology: localization, periodicity and Galois symmetry的笔记。同年，他受邀在法国尼斯举行的国际数学家大会上作报告，作了题为Galois symmetry in manifold theory at the primes的演讲。这项工作的重要性可以从以下事实清楚地看出：2005年，即写成三十五年后，讲稿笔记和苏利文在尼斯大会上的演讲均由Springer-Verlag出版。弗瑞兹·约翰 McCleary在评论2005年出版物[62]时写道：-

1970年，苏利文散发了一套笔记，即“MIT笔记”，将拓扑空间的局部化和完备化引入同伦论，以及其他对拓扑学发展产生重大影响的重要概念和构造。该笔记的一个版本以“同伦论的遗传学和Adams猜想”（1974）为题发表。尽管自1970年以来已过去很长时间，它们现在的出版不仅仅是一项历史性工作。……C T C Wall在谈到MIT笔记时写道：“很难如此简短地总结苏利文的工作：应当阅读（笔记中）完整的哲学阐述。”笔记中的阐述集中于认识论问题——特别是，流形的底层代数本质是什么，我们如何知道它？……我旧的MIT笔记副本中包括作者的一张照片，经过如此多的复印后几乎无法辨认。新书中作者及其子女的照片以及后记，为了解深刻数学思想的发展提供了难得的洞见。这些笔记仍然值得一读，因为其思想的大胆、对所掌握现有结构的清晰驾驭，以及它们为几何拓扑提供的新图景。必须感谢编者[Andrew Ranicki]使这些笔记能为另一代拓扑学家所用。

在伯克利，一座新建筑Evans Hall于1971年竣工，供数学系使用。许多人认为其设计绝对难看，研究生们决定粉刷一些墙壁以照亮他们的环境。苏利文在2002年写信给Lee Mosher，解释了他所起的作用[40]：——

1971年，我作为加利福尼亚大学的客人，在数学系讲课。与此同时，校董与研究生等人之间发生了对抗。后者计划继续通过绘制有吸引力的壁画来装饰系里的墙壁，而校董禁止了此事。喝茶时，一些学生走过来，邀请我第二天加入他们的绘画。当一个留胡子的家伙[威廉·瑟斯顿]给我看了一幅令人难以置信的图，画的是三孔圆盘中嵌入的曲线，并问我是否认为这画起来会很有趣时，我变得热情起来。我说：“当然，”第二天我们花了一整个下午来做这件事。当我们把图形转移到墙上时，很自然且自动地一次处理一束束的线——作为一个近似的叶状结构——然后在最后只要数目对得上就把它们连接起来。因此几年后，在76年，当Bill就他的曲面变换理论作了一场即兴的3小时演讲时，在经历了71年几个小时的绘画之后，我毫不费力地在启发式层面上吸收了它。

苏利文在1973-74学年在法国访问巴黎-奥赛大学。在这次逗留结束时，他受邀成为巴黎郊外高等科学研究所的永久教授，并于1974年就任此职。在IHÉS，苏利文[33]：——

……是一位善于在来访者中组织活动、激发兴趣的大师，对年轻人尤其有效。最近的约翰·查尔斯·菲尔兹奖得主柯蒂斯·麦克马伦就是苏利文影响力的一个很好的例子：尽管柯蒂斯·麦克马伦在哈佛大学获得博士学位，他实际上是苏利文的学生，正是在访问IHÉS期间，柯蒂斯·麦克马伦产生了他的学位论文问题的想法。另一个例子是米哈伊尔·格罗莫夫：正是苏利文的邀请，才在1977年首次把米哈伊尔·格罗莫夫作为访问者带到IHÉS，那是在米哈伊尔·格罗莫夫离开苏联三年之后。

1981年，苏利文被任命为纽约市立大学研究生中心的阿尔伯特·爱因斯坦科学讲席，但继续以兼职方式在高等科学研究所[29]保留他的教授职位：-

在20世纪80年代，[阿尔伯特·爱因斯坦]讲席的资源使得能够创立一个关于几何与混沌理论的定期讨论班，把一流的国际学者带到CUNY[纽约市立大学]和纽约市。随后，该讨论班一直得到研究生中心的支持，探究拓扑学与由量子场论和流体力学所提供的自然数学模型之间的联系。

在写到这个讨论班时，苏利文解释了代数拓扑与量子场论之间的紧密关系从何而来[29]：-

我尤其感兴趣的是代数拓扑的方法，它把线性对象（同调群）与带点的非线性对象（流形……）联系起来，就像量子理论把状态线性空间与带点的经典系统联系起来一样。代数拓扑中的主要角色是幂零算子或边界算子，而在量子场论中，被称为Q和“delta”的幂零算子扮演着重要角色，它们编码特定理论的作用量中存在的任何对称性，并衡量对所有路径的积分不变地赋予意义的障碍。在代数拓扑中有一个强有力的思想，最初归功于Stasheff，但超越了他著名的、优雅的无穷同伦结合代数概念，它允许人们在一个新的世界中容忍略微错误的代数恒等式，在那里它们变得有效为真。在量子场论中，正则化或截断的必要性有时会破坏——但只是略微破坏——表达各种对称性和结构的恒等式，这可能提供一个机会来使用这个来自代数拓扑的强有力的思想。最后，代数与几何拓扑一直把它的努力指向以代数方式理解像流形这样的几何对象，流形是时空的经典模型，而量子场论往往从定义在遍布时空的经典场上的经典作用量开始指定一个特定理论，然后进行其代数算法。

1996年，苏利文辞去他在巴黎的教授职位，去担任纽约州立大学石溪分校的数学教授，同时继续在纽约市立大学研究生中心保留他的兼职职位。在1998-99学年，SUNY把苏利文提升为杰出教授。他们的公告如下[1]：-

苏利文，石溪数学系，是我们时代最伟大的数学家之一，也是过去100年里最重要的拓扑学家之一。他对数学的多个不同领域做出了贡献，包括拓扑学、几何学、动力系统和复分析。

苏利文还获得了其他重要荣誉。除了上述1971年奥斯瓦尔德·维布伦几何学奖之外，他还于1981年获得了法国科学院颁发的埃利·嘉当奖，1994年获得了费萨尔国王国际科学奖（数学），并于1998年获得了巴西科学院颁发的国家科学勋章。他于1997年获得了纽约市市长科学与技术卓越奖。2004年，乔治·W·布什总统在白宫举行的仪式上授予他国家科学奖章[35]：-

因其在数学方面的成就，包括解决了一些最困难的问题并创建了全新的活动领域，以及揭示了看似无关领域之间惊人而出人意料的联系。

导致他获得该奖章贡献的描述见[35]：-

苏利文的早期工作是在同伦论和割补术方面，他为此带来了新的几何观点。他的几何洞察力导致了关于流形拓扑学的许多重要结果。他基于微分形式的实同伦型和有理同伦型理论有着深刻的应用，例如，在复代数簇的拓扑学中。苏利文在叶状结构和动力系统的研究中做出了重要贡献。他还证明了拟共形和鲁道夫·利普希茨流形的基本结果，这些范畴介于拓扑流形和光滑流形之间。在20世纪80年代和90年代，他促成了共形动力系统领域的出现，使其成为跨越纯数学和应用数学传统边界的活跃而重要的数学分支。近年来，他开创了弦拓扑学领域。

2006年，苏利文获得了美国数学会颁发的Leroy P 凯瑟琳·斯蒂尔终身成就奖。引文写道[2]：-

苏利文 对数学的许多分支做出了根本性的贡献。苏利文 的局部化理论和 埃瓦里斯特·伽罗瓦 对称性理论，在他1970年著名的麻省理工学院[Massachusetts Institute of Technology]讲义中传播开来，一直是同伦论中许多后续发展的核心。苏利文 用它解决了组合流形的 弗兰克·亚当斯 猜想和主猜想。后来 苏利文 发展并应用有理同伦论来解决关于闭测地线、有限复形的自同构群、埃里希·凯勒 流形的拓扑学以及光滑流形分类的问题。他多次重塑自己，在动力系统、克莱因群和低维拓扑学中扮演了主要或主导角色。这些简短的评论不足以体现 苏利文 思想和影响的广度。除了他发展的具体理论和解决的问题——这里还有许多重要的未提及——他对数学的统一愿景渗透在他的工作中，并激励了他周围的人。多年来，他一直是 IHÉS [Institut des Hautes Études Scientifiques] 数学对话的中心。后来他搬到纽约，他的每周讨论班仍然是该市数学生活的一个重要特色。

他因对代数拓扑学和共形动力学的贡献于2010年获得沃尔夫数学奖 [60]：-

苏利文 在许多领域做出了根本性的贡献，特别是在代数拓扑学和动力系统方面。他的早期工作为高维流形分类的割补理论方法奠定了基础，尤其是提供了给定同伦型内单连通流形的完全分类。他发展了同伦论中的局部化和完备化概念，并利用这一理论证明了 Adams 猜想（也由 丹尼尔·奎伦 独立证明）。苏利文 和 丹尼尔·奎伦 引入了空间的有理同伦型。苏利文 表明可以使用相关微分分次代数的最小模型来计算它。苏利文 的思想在代数拓扑学中产生了深远的影响和应用。苏利文 最重要的贡献之一是锻造了新的数学技术，以严格确立 米切尔·费根鲍姆 的重整化作为动力系统中普适性现象解释的预测。苏利文 的“无游荡域”定理解决了 波恩哈德·黎曼 球面上迭代有理映射的动力系统分类，解决了 皮埃尔·法图 和 茹利亚 六十年前的猜想。他的工作通过将拟共形方法引入该领域，并在有理映射和克莱因群之间建立了一个持续令人感兴趣的启发式词典，引发了大量活动。他的克莱因群刚性定理在 奥斯瓦尔德·泰希米勒 理论和 威廉·瑟斯顿 的3-流形几何化纲领中有重要应用。他最近关于拓扑场论和弦论形式体系的工作可以看作是他寻求对空间本质及其如何编码在奇异代数结构中的终极理解的副产品。苏利文 的工作一直具有创新性和启发性。除了解决困难的未决问题外，他的工作还产生了许多数学家追求的重要而活跃的研究领域。

苏利文 继续被授予最负盛名的奖项。2014年，他被授予巴尔赞数学奖（纯数学/应用数学）[15]：-

因他对拓扑学和动力系统理论的重大贡献，为后代开辟了新的视角。因他在数学许多领域（如拓扑学、几何学、克莱因群理论、分析和数论）的杰出成果。

他当选为 国家科学院 成员（1983年）、Brazilian National Academy of Sciences 成员（1984年）和 纽约科学院 成员（1984年）。他还当选为 美国艺术与科学院 会士（1991年）、Irish Royal Academy 通讯成员（2012年）和 伦敦数学会 荣誉成员（2013年）。他曾担任 美国数学会 副主席（1990-93年）。他获得了华威大学（1983年）和里昂高等师范学院（2001年）的荣誉学位。另一项荣誉是2002年9月在纽约市立大学研究生中心举行的会议，以庆祝他被任命为纽约市立大学 阿尔伯特·爱因斯坦 科学讲席20周年。他于2012年当选为 美国数学会 会士。

尼尔斯·阿贝尔奖被公认为授予数学家的最高奖项。该奖于2022年颁发给苏利文[63]：

……因其在最广泛意义上的拓扑学，特别是其代数、几何和动力方面做出的开创性贡献。

关于该奖项的新闻稿和颁奖词，见THIS LINK。

苏利文现在与数学家Moira Chas结婚。她于1998年获得巴塞罗那自治大学的博士学位，学位论文为Minimum Periods of Homeomorphisms of Orientable Surfaces。他有六个孩子：Lori、Amanda、Michael、Tom、Ricardo和Clara。Michael Sullivan是一位数学家，于1999年获得斯坦福大学的博士学位，学位论文为K-theoretic invariants for Floer homology。

最后，让我们先引用苏利文在采访[26]中的一些想法：-

我可以讲一些我对研究生说的话：批判性思维很重要。批判性地思考、审视你的信念、理解它们为何被普遍持有，这很好，然后也许在某些情况下，你必须稍作修改，使它们更好地发挥作用。这就是帮助我更好地理解数学的原因。例如，甚至你从硕士阶段学到的东西，有时也是他们的视角。拥有一个视角是极好的，这有点像一种偏见。它是好的，因为它使你更有效率，你可以把精力投入到那些方向上，对吧？但有时，它并不正确。在某些情况下或某些观点下，有另一种看待它的方式。这可能帮助你在被先前视角阻塞的方向上取得进展。这不是消极意义上的批判，而是审视意义上的批判。所以批判性思维是，我借用伯特兰·罗素在1952年一次精彩采访中的话，他说了许多非常迷人和非常聪明的事情，但他也强调了这一点：当你有一个视角时，它有时允许你做出非理性的理性决定。所以保持批判是好的，甚至对你自己的信念也是如此，因为它有帮助。这在数学中也同样适用。

最后引用他在[64]中的想法：-

数学这个词名声不好。人人都说“我很不擅长数学”，但这很荒谬。当他们五六岁的时候，他们做得很好；他们在探索空间，充满好奇。坚持成为数学家的人们只是那些克服了困难并且仍然对新发现感兴趣的人，因为这很迷人。例如，幼儿并不是在玩耍，他们在工作，他们在弄清楚事物是如何运作的。首先，他们研究空间，然后研究数字——他们正在起步，他们是小数学家。

Dennis Parnell Sullivan should really be Dennis Parnell Sullivan III since both his father and grandfather were named Dennis Parnell Sullivan. His father, known as Dennis Parnell Sullivan Jr (1920-1978), was born in Port Huron, Michigan, on 27 July 1920. He worked for the Hexagon Tool and Engineering Company in Dearborn, Wayne, Michigan. Dennis Parnell Sullivan Jr married Rita Jane Treend (1922-2014) in Port Huron, St Clair, Michigan on 25 March 1940. Rita was born on 16 June 1922, the daughter of Charles G Treend, the manager of a Department Store, and his wife Frances. Dennis and Rita Sullivan had two children, Dennis Parnell Sullivan (born 12 February 1941, the subject of this biography) and Michael Francis Sullivan (born 22 July 1942). Michael died on 26 July 1957 in Houston, Texas, at the age of fifteen after a long medical battle. Rita Sullivan filed for divorce on 11 September 1945 and it was granted absolute on 7 August 1946.

Although Dennis Sullivan was born in Port Huron, Michigan, he was brought up by his mother in Houston, Texas, and has always considered himself a Texan. When his mother died in 2014, her obituary stated [48]:-

Rita J Sullivan a long time Houstonian whose career spanned radio, television, advertising and real estate died Saturday morning August 9, 2014.

He attended school in Houston before entering Rice University, also in Houston, to study chemistry. He explained in [58] that he applied to Rice University despite being advised not to by his high school teacher:-

I was sort of a bad student in high school, in the sense that I didn't study. I think it's because I was actually near-sighted. I didn't know you were supposed to be able to see the blackboard. So when I told my teacher that I was applying to Rice University, which is a very good school right in Houston, Texas, where I lived, she said, "You shouldn't apply there. You're not a good enough student."

He switched from chemistry to mathematics when he found that the mathematics courses he took were the most exciting of all the science courses [58]:-

We took a lot of science. I didn't know there were such things as mathematicians. And then, in my second-year math course, the professor, [the late] Guy Johnson, was showing us a wonderful theorem in math that went way beyond the intellectual quality of things like equations and calculations, which is what I thought math was. And I was just very struck by it. Once you understand that deeper real mathematics, it's sort of amazingly clear what things mean. There's no ambiguity, whereas almost everything else is much woollier and murkier, compared with mathematics. There's some great comfort in that. That's what attracted me.

The professor that turned Sullivan into a mathematician was Guy Johnson Jr (1922-2017). He had been awarded a Ph.D. from Rice University in 1955 for his 2-part thesis Part I: Collective Singularities of Families of Analytic Functions. Part II: Regions of Flatness for Analytic Functions and their Derivatives written with Szolem Mandelbrojt as his thesis advisor. Since this event was so crucial, we give more details. Guy Johnson [26]:-

... drew a picture of a kidney-shaped swimming pool, and a round swimming pool. And he said, you could deform this kidney-shaped swimming pool into the round one. At each point, the distortion is by scaling. A little triangle at this point goes to a similar triangle at the other point. At every point, that's true. We have a formula for the mapping, because we're taking calculus, and we had a notation for discussing it, which we have been studying. But this was like a geometric picture. This mapping was essentially unique. And this was, the nature of this statement was totally different from any math statement I've ever seen before. It was, like, general, deep, and wow! And true! So then, within a few weeks, I changed my major to math.

There was a lesson he learnt while a mathematics major at Rice University which he felt was important for his approach to research [64]:-

... we had a homework assignment that had one problem on it that no one could do so I just kept working on it for some reason. It was a very strange problem, and after two weeks I got it. But I wasn't the best student in the class, there were several really good students. It was then that I realised that if you really want to understand something, you have to keep plugging at it and you'll eventually understand it, and that's basically it. So then you have to find simple things that you can work with so you're not totally confused by complexity and information.

Sullivan was awarded a B.A. by Rice in 1963. He married Kathleen Rose McGuire on 10 August 1963 in Harris, Texas. He then went to Princeton University to undertake graduate studies in mathematics. At Princeton his thesis advisor was William Browder and Sullivan was awarded a Ph.D. in 1966 for his thesis Triangulating Homotopy Equivalences. Anthony Phillips writes [44]:-

William Browder's students at that time were mining surgery theory for topological gold. Dennis' thesis was in this vein and led to his work on the Hauptvermutung (1967).

It was this work which led to Sullivan receiving the Oswald Veblen Prize in Geometry in 1971 from the American Mathematical Society. The sixth award of this prize was made [61]:-

To Dennis P Sullivan for his work on the Hauptvermutung summarised in the paper 'On the Hauptvermutung for manifolds', Bulletin of the American Mathematical Society, volume 73 (1967) ...

After the award of his doctorate Sullivan was appointed to a NATO Fellowship at Warwick University in England. I [EFR] was a research student at Warwick at this time and I remember many evenings when I played bridge with fellow research students while Dennis worked at an adjacent table writing mathematics papers. After holding the fellowship at Warwick, Sullivan held a Miller Fellowship at the University of California at Berkeley working on the Adams conjecture, K-theory, and étale homotopy. In 1969 he went to the Massachusetts Institute of Technology as a Sloan Fellow of Mathematics where [40]:-

... his work focused on what he named geometric topology (in particular the study of Galois symmetry) and on the construction of minimal models for the rational homotopy type of manifolds, using differential forms.

In 1970 he produced a set of notes entitled Geometric topology: localization, periodicity and Galois symmetry. In the same year he was an invited speaker at the International Congress of Mathematicians in Nice, France, where he gave the lecture Galois symmetry in manifold theory at the primes. The importance of this work can be clearly seen from the fact that in 2005, thirty-five years after they were written, both the lecture notes and Sullivan's lecture to the Nice Congress were published by Springer-Verlag. John McCleary writes in a review of the 2005 publication [62]:-

In 1970, Sullivan circulated a set of notes, the "MIT notes", introducing localisation and completion of topological spaces to homotopy theory, and other important concepts and constructions that have had a major influence on the development of topology. A version of the notes appeared as "Genetics of homotopy theory and the Adams conjecture" (1974). Although it has been a long time since 1970, their publication now is more than an historical exercise. ... C T C Wall wrote of the MIT notes, "it is difficult to summarise Sullivan's work so briefly: the full philosophical exposition in (the notes) should be read." The exposition in the notes focuses on epistemological questions - in particular, what is the underlying algebraic nature of a manifold, and how can we know it? ... My old copy of the MIT notes included a photo of the author, barely recognisable after so much photocopying. The photos in the new book of the author and his children together with the Postscript give a rare insight into the development of deep mathematical ideas. The notes remain worth reading for the boldness of their ideas, the clear mastery of available structure they command, and the fresh picture they provide for geometric topology. The editor [Andrew Ranicki] must be thanked for making the notes available to another generation of topologists.

At Berkeley a new building, Evans Hall, was completed in 1971 to house the Mathematics Department. Many thought it a decidedly unattractive design and the postgraduates decided to paint some walls to brighten up their environment. Sullivan wrote to Lee Mosher in 2002 explaining the part he played [40]:-

In 1971 I was a guest of the University of California giving lectures in the Math Dept. At the same time there was a confrontation between the trustees and the graduate students et al. The latter planned to continue decorating the walls of the department by painting attractive murals and the trustees forbade it. At tea some students came up and invited me to join their painting the next day. I became enthusiastic when one bearded fellow [Bill Thurston] showed me an incredible drawing of an embedded curve in the triply punctured disk and asked if I thought this would be interesting to paint. I said, 'You bet,' and the next day we spent all afternoon doing it. As we transferred the figure to the wall it was natural and automatic to do it in terms of bunches of strands at a time - as an approximate foliation - and then connect them up at the end as long as the numbers worked out. Thus some years later in '76 when Bill gave an impromptu 3-hour lecture about his theory of surface transformations I absorbed it painlessly at a heuristic level after the experience of several hours of painting in '71.

Sullivan spent the academic year 1973-74 in France visiting the University of Paris-Orsay. At the end of this stay he was invited to become a permanent professor at the Institut des Hautes Études Scientifiques outside Paris and he took up this position in 1974. At the IHÉS, Sullivan [33]:-

... was a master at orchestrating activity and interest among visitors and was especially effective with young people. Recent Fields Medalist Curtis McMullen is a good example of Sullivan's influence: Although McMullen received his Ph.D. from Harvard, he was really Sullivan's student, and it was while visiting the IHÉS that McMullen got the idea for his thesis problem. Another example is Gromov: It was Sullivan's invitation that first brought Gromov to the IHÉS as a visitor in 1977, three years after Gromov had gotten out of the Soviet Union.

In 1981 Sullivan was appointed to the Albert Einstein Chair in Science at the Graduate Center of the City University of New York but continued to hold his professorship on a part-time basis at the Institut des Hautes Études Scientifiques [29]:-

During the 1980s the resources of the [Albert Einstein] Chair allowed the founding of a regular seminar in geometry and chaos theory that brought first-rank international scholars to CUNY [the City University of New York] and New York City. Subsequently, the seminar has been supported by The Graduate Center, pursuing the connections between topology and the mathematical models of nature provided by quantum field theory and fluid mechanics.

Writing about this seminar, Sullivan explained where the strong relationship between algebraic topology and quantum field theory arises [29]:-

I am particularly interested in the method of algebraic topology which associates linear objects (homology groups) to nonlinear objects with points ( manifolds...) just like quantum theory associates linear spaces of states to classical systems with points. The main character in algebraic topology is the nilpotent operator or boundary operator while in quantum field theory an important role is played by the nilpotent operators called Q and "delta" which encode whatever symmetry is present in the action of the particular theory and measure the obstruction to invariantly assign meaning to the integral over all paths. In algebraic topology there is a powerful idea, due first to Stasheff but going beyond his famous and elegant concept of an infinitely homotopy associative algebra, which allows one to live with slightly false algebraic identities in a new world where they become effectively true. In quantum field theory the necessity to regularise or cutoff which sometimes destroys, but only slightly, identities expressing various symmetries and structures may provide an opportunity to use this powerful idea from algebraic topology. Finally algebraic and geometric topology has always directed it efforts towards understanding in an algebraic way geometric objects like manifolds which are the classical models of spacetime, while quantum field theory often begins its specification of a particular theory with the classical action defined on the classical fields spread over spacetime and then proceeds to its algebraic algorithms.

In 1996 Sullivan resigned from his professorship in Paris to take up a professorship in mathematics at the State University of New York at Stony Brook, continuing to hold his part-time position at the Graduate Center of the City University of New York. In session 1998-99 SUNY promoted Sullivan to Distinguished Professor. Their announcement reads as follows [1]:-

Dennis Parnell Sullivan, Department of Mathematics, Stony Brook, is one of the great mathematicians of our time and one of the most important topologists of the last 100 years. He has contributed to several diverse areas of mathematics including topology, geometry and dynamics and complex analysis.

Sullivan has received other major honours. In addition to the 1971 Oswald Veblen Prize in Geometry mentioned above, he received the 1981 Prix Élie Cartan from the French Academy of Sciences, the 1994 King Faisal International Prize for Science (mathematics), and the Ordem Scientifico Nacional by the Brazilian Academy of Sciences in 1998. He received the New York City Mayor's Award for Excellence in Science and Technology in 1997. He was awarded the 2004 National Medal of Science by President George W Bush at a ceremony in the White House [35]:-

For his achievements in mathematics, including solving some of the most difficult problems and creating entirely new areas of activity, and for uncovering striking, unexpected connections between seemingly unrelated fields.

The description of his contributions leading to the award of the Medal are described in [35]:-

Sullivan's early work was in homotopy theory and surgery, to which he brought a new, geometric point of view. His geometric insights led to many important results on the topology of manifolds. His theory of real and rational homotopy types, based on differential forms, has had profound applications, for example, to the topology of complex algebraic varieties. Sullivan has made important contributions to the study of foliations and dynamical systems. He has also proved foundational results on quasiconformal and Lipschitz manifolds, categories that are intermediate between the topological and smooth ones. During the 1980s and 1990s, he was responsible for the emergence of the field of conformal dynamics as a lively and important branch of mathematics straddling the traditional borders between pure and applied areas. In recent years, he launched the field of string topology.

In 2006 Sullivan received the Leroy P Steele Prize for Lifetime Achievement from the American Mathematical Society. The citation states [2]:-

Dennis Sullivan has made fundamental contributions to many branches of mathematics. Sullivan's theory of localisation and Galois symmetry, propagated in his famous 1970 MIT [Massachusetts Institute of Technology] notes, has been at the heart of many subsequent developments in homotopy theory. Sullivan used it to solve the Adams Conjecture and the Hauptvermutung for combinatorial manifolds. Later Sullivan developed and applied rational homotopy theory to problems about closed geodesics, the automorphism group of a finite complex, the topology of Kähler manifolds, and the classification of smooth manifolds. He has reinvented himself several times, playing major or dominant roles in dynamical systems, Kleinian groups, and low dimensional topology. These brief remarks do not do justice to the scope of Sullivan's ideas and influence. Beyond the specific theories he has developed and the problems he has solved - and there are many significant ones not mentioned here - his uniform vision of mathematics permeates his work and has inspired those around him. For many years he was at the centre of the mathematical conversation at IHÉS [Institut des Hautes Études Scientifiques]. Later he moved to New York where his weekly seminar remains an important feature of mathematical life in the City.

He received the Wolf Prize in Mathematics in 2010 for his contributions to algebraic topology and conformal dynamics [60]:-

Dennis Sullivan has made fundamental contributions in many areas, especially in algebraic topology and dynamical systems. His early work helped lay the foundations for the surgery theory approach to the classification of higher dimensional manifolds, most particularly providing a complete classification of simply connected manifolds within a given homotopy type. He developed the notions of localisation and completion in homotopy theory and used this theory to prove the Adams conjecture (also proved independently by Quillen). Sullivan and Quillen introduced the rational homotopy type of space. Sullivan showed that it can be computed using a minimal model of an associated differential graded algebra. Sullivan's ideas have had far-reaching influence and applications in algebraic topology. One of Sullivan's most important contributions was to forge the new mathematical techniques needed to rigorously establish the predictions of Feigenbaum's renormalisation as an explanation of the phenomenon of universality in dynamical systems. Sullivan's "no wandering domains" theorem settled the classification of dynamics for iterated rational maps of the Riemann sphere, solving a sixty-year-old conjecture by Fatou and Julia. His work generated a surge of activity by introducing quasiconformal methods to the field and establishing an inspiring dictionary between rational maps and Kleinian groups of continuing interest. His rigidity theorem for Kleinian groups has important applications in Teichmüller theory and for Thurston's geometrisation programme for 3-manifolds. His recent work on topological field theories and the formalism of string theory can be viewed as a by-product of his quest for an ultimate understanding of the nature of space and how it can be encoded in strange algebraic structures. Sullivan's work has been consistently innovative and inspirational. Beyond the solution of difficult outstanding problems, his work has generated important and active areas of research pursued by many mathematicians.

Sullivan continued to be awarded the most prestigious prizes. In 2014 he was awarded the Balzan Prize for Mathematics (Pure/Applied) [15]:-

For his major contributions to topology and the theory of dynamical systems, opening new perspectives for generations to come. For his exceptional results in many fields of mathematics, such as topology, geometry, the theory of Kleinian groups, analysis, and number theory.

He was elected a member of the National Academy of Sciences (1983), of the Brazilian National Academy of Sciences (1984), and of the New York Academy of Sciences (1984). He was also elected a fellow of the American Academy of Arts and Sciences (1991), a corresponding member of the Irish Royal Academy (2012) and an honorary member of the London Mathematical Society (2013). He has served the American Mathematical Society as vice-president (1990-93). He received honorary degrees from the University of Warwick (1983) and the École Normale Supérieure de Lyon (2001). Another honour was the conference held at the CUNY Graduate Center in September 2002 to celebrate the 20th anniversary of his appointment as the Albert Einstein Chair in Science at The City University of New York. He was elected a fellow of the American Mathematical Society in 2012.

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented to Sullivan in 2022 [63]:

... for his ground-breaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects.

For the Press Release and the Citation for this award, see THIS LINK.

Sullivan is now married to the mathematician Moira Chas. She was awarded a Ph.D. from the Universidad Autónoma de Barcelona in 1998 for her thesis Minimum Periods of Homeomorphisms of Orientable Surfaces. He has six children: Lori, Amanda, Michael, Tom, Ricardo and Clara. Michael Sullivan is a mathematician with a Ph.D. from Stanford University in 1999 for his thesis K-theoretic invariants for Floer homology.

Let us end by first quoting some thoughts from Sullivan from the interview [26]:-

I could say something that I say to my graduate students: critical thinking is important. It's good to think critically, examine your beliefs, understand why they're commonly held, and then maybe, in certain circumstances, you have to modify them slightly, to make them work better. That's what has helped me understand mathematics better. For example, even what you learned from your masters, sometimes, it's their perspective. Having a perspective is excellent, which is kind of like a bias. It is good because it makes you more effective and you can put your energy in those directions, right? But then sometimes, it's not right. In some situations or some points of view, there's a different way to look at it. And this may help you make progress in a direction that was blocked with the previous perspective. This is not being critical in the sense of being negative, it's critical in the sense of examining. So critical thinking is, and I'm borrowing this from a wonderful interview of Bertrand Russell in 1952, he says a lot of very charming and very intelligent things, but he also emphasises this point that when you have a perspective, it sometimes allows you to make irrational rational decisions. So it's good to be critical, even of your own beliefs because it helps. That works in mathematics too.

Finally quoting his thoughts from [64]:-

The word mathematics has a bad reputation. Everyone says "I'm very bad at mathematics" but that's ridiculous. When they were five or six years old they were fine; they were exploring space and being curious. People who stay mathematicians are just the ones that got past the difficulties and are still interested in new discoveries because it is fascinating. Toddlers, for instance, aren't really playing, they're working, they're figuring out how things work. First, they do space and then they do numbers - they're starting, they're little mathematicians.