迈克尔·弗里德曼Michael Freedman

迈克尔·弗里德曼的父母Benedict和Nancy 弗里德曼都相当有名。他的母亲Nancy Mars于1920年出生于芝加哥，在进入芝加哥艺术学院、洛杉矶城市学院和南加州大学之前曾出演过一些角色。她于1941年6月29日与Benedict Freedman结婚。Benedict Freedman的父亲弗洛伦斯·南丁格尔·大卫出生于罗马尼亚。Benedict是一位才华横溢的数学家、音乐家和作家。他曾就读于加利福尼亚州格伦代尔的柯蒂斯-赖特技术学院，获得航空工程学位。20世纪40年代，他在柯蒂斯-赖特教授航空工程，同时还有一份平行的职业，担任广播节目编剧、戏剧评论家和报纸编辑。Benedict和Nancy 弗里德曼是几部著名小说的合著者。他们有三个孩子：Johanna、弗里德曼和Deborah。
弗里德曼在成长过程中展现出非凡的数学天赋。然而，他也喜欢以表现主义风格绘画，1968年进入加利福尼亚大学乔治·伯克利分校时，他仍未在数学和艺术之间做出决定。他很快就坚定地决定学习数学，但在第一学年临近结束时，他决定申请到普林斯顿大学攻读研究生。弗里德曼喜欢下围棋，他知道普林斯顿的数学家拉尔夫·福克斯是一位围棋冠军。他读了拉尔夫·福克斯1963年的著作Introduction to Knot Theory，并在1969年申请普林斯顿时附上了自己的猜想。1973年，他因博士论文Codimension-Two Surgery被普林斯顿授予博士学位。他的学位论文导师是威廉·布劳德。

毕业后，弗里德曼被任命为加利福尼亚大学伯克利分校数学系讲师。他从1973年起担任此职，直到1975年成为普林斯顿高等爱德华·斯图迪研究所成员。1976年，他被任命为加利福尼亚大学圣迭戈分校数学系助理教授。

弗里德曼于1979年在圣迭戈晋升为副教授。1980/81学年，他在普林斯顿高等斯图迪研究所度过，之后回到加利福尼亚大学圣迭戈分校，并于1982年晋升为教授。除这一职位外，他还于1985年被任命为Charles Lee Powell数学讲席。

弗里德曼因在儒勒·昂利·庞加莱猜想方面的工作于1986年获得菲尔兹奖。儒勒·昂利·庞加莱猜想是20世纪数学著名问题之一，它断言一个单连通的闭3维流形是一个3维球面。高维儒勒·昂利·庞加莱猜想声称，任何与$n$球面homotopy equivalent的闭$n$-流形必定是$n$-球面。当$n = 3$时，这等价于儒勒·昂利·庞加莱猜想。斯蒂芬·斯梅尔于1961年对至少为5的$n$证明了高维儒勒·昂利·庞加莱猜想。弗里德曼于1982年对$n = 4$证明了该猜想，但原猜想一直悬而未决，直到被格里戈里·佩雷尔曼解决，他因这一证明被授予2006年约翰·查尔斯·菲尔兹奖章。

约翰·米尔诺在描述弗里德曼的工作时——该工作使他在1986年于伯克利举行的国际数学家大会上获得菲尔兹奖章——说道：-

弗里德曼不仅证明了4维topological流形的儒勒·昂利·庞加莱猜想，从而刻画了球面$S^{4}$，而且还为我们给出了更一般的4维流形的分类定理，这些定理易于陈述和使用，但难以证明。他在拓扑情形下结果的简单性质，必须与如今已知在研究可微和分段线性4维流形时出现的极端复杂性形成对比。……弗里德曼1982年对4维儒勒·昂利·庞加莱猜想的证明是一项非凡的壮举。他的方法如此犀利，以至于实际上给出了所有紧致单连通拓扑4维流形的完整分类，产生了许多此前未知的此类流形实例，以及许多此前未知的已知流形之间的同胚。

弗里德曼因其工作获得了许多荣誉。他于1984年被评为加州年度科学家，同年，他被授予麦克阿瑟基金会研究员称号，并当选为国家科学院院士。1985年，他当选为美国艺术与科学院院士。除了在1986年获得菲尔兹奖章外，他还在同年获得了美国数学会颁发的奥斯瓦尔德·维布伦奖。奥斯瓦尔德·维布伦奖的颁奖词写道（见[3]）：-

在60年代初发现了儒勒·昂利·庞加莱猜想的证明以及维数大于四的单连通流形的其他性质之后，最大的未解决问题之一，除了三维儒勒·昂利·庞加莱猜想之外，就是闭单连通四维流形的分类。在其发表于《微分几何杂志》（1982年）的论文《四维流形的拓扑》中，弗里德曼解决了这个问题，特别是四维儒勒·昂利·庞加莱猜想。主要创新在于通过证明Casson提出的一个同伦论条件——用于在带边四维流形中嵌入一个2-柄，即一个加厚圆盘——解决了单连通手术问题。

除了这些关于闭单连通四维流形的结果外，弗里德曼还证明了：

1. 任何与$\mathbb{R}^{4}$真等价的四维流形都同胚于$\mathbb{R}^{4}$；对于$S^{3} \times \mathbb{R}$也有一个相关的结果。
2. 存在不可光滑化的闭四维流形。
3. 四维的Hauptvermutung是假的；即存在具有不等价组合三角剖分的四维流形。

最后，我们注意到上述论文的结果，连同西蒙·唐纳森的工作，产生了$\mathbb{R}^{4}$的奇异光滑化的惊人例子。

在他的回复中，弗里德曼感谢了他的老师们（他说其中包括他的学生），并给出了一些关于数学的引人入胜的观点[3]：

我对几何学的主要兴趣在于它揭示流形拓扑学的方式。在这里，似乎重要的是对从形式到具体的整个几何谱系持开放态度。所谓谱系，我指的是我们思考数学结构的各种方式。在一个极端，问题的直觉几乎完全来自心理图像。在另一个极端，几何负担被转移到符号和代数思维上。当然，从代数的观点来看，这个极端只是一个中间地带，代数准备在形式运算的方向上走得更远，并完全放弃几何直觉。

在同一回复中，弗里德曼还谈到了数学对世界可能产生的影响，以及数学家应如何表达他们的想法：-

在十九世纪，有一场运动，雅各布·施泰纳是其主要代表，旨在保持几何学的纯粹性，抵御代数的侵蚀。今天我认为我们感到数学的许多力量来自于结合该学科看似遥远分支的洞见。数学与其说是一组不同的科目，不如说是一种思维方式。因此，它可以应用于任何知识分支。我要赞扬数学家们现在为发表关于教育、能源、经济、国防和世界和平的想法所做的努力。数学内部的经验表明，不必是某个领域的老手才能做出贡献。在数学之外，情况不太清楚，但我不禁感到，在那里，将重要问题完全留给专家也是一个错误。

1987年6月，弗里德曼在白宫由罗纳德·里根总统授予国家科学奖章。次年，他获得了洪堡奖，并于1994年获得了古根海姆奖学金奖。

弗里德曼继续在加州大学圣迭戈分校担任查尔斯·李·鲍威尔数学讲席教授，直到1998年他离开学术界，接受了Station Q的职位，这是微软的一个研究小组，从事拓扑量子计算研究。这样一来，弗里德曼成为第一位离开学术界为公司工作的菲尔兹奖章得主。弗里德曼成为Station Q的主任，为了了解他研究的一些主题，我们将看看他在随后十年里所写论文的一些标题。首先我们注意到，他于1998年在柏林国际数学家大会上就Topological views on computational complexity作了邀请报告。然后他发表了诸如：Quantum computation and the localization of modular functors（2001）；Projective plane and planar quantum codes（2001）；Poly-locality in quantum computing（2002）；Simulation of topological field theories by quantum computers（2002）；Topological quantum computation (2003); Approximate Counting and Quantum Computation（2005）；Topological quantum computation（2006）；Topological quantum computing with only one mobile quasiparticle（2006）；Interacting anyons in topological quantum liquids: The golden chain（2008）；Measurement-Only Topological Quantum Computation（2008）；以及Topological Phase in a Quantum Gravity Model（2008）等论文。他还发表了一些与量子计算无关的论文，如Extension of incompressible surfaces on the boundaries of 3-manifolds（2000）、Diameters of Homogeneous Spaces（2003）和Covering a nontaming knot by the unlink（2007）。

Michael Freedman's parents Benedict and Nancy Freedman are both quite famous. His mother, Nancy Mars, was born in Chicago in 1920 and had some acting roles before attending the Chicago Art Institute, Los Angeles City College, and the University of Southern California. She married Benedict Freedman on 29 June 1941. Benedict Freedman's father, David, was born in Romania. Benedict was a talented mathematician, musician and writer. He studied at Curtiss-Wright Technical Institute at Glendale, California, graduating with a degree in aeronautical engineering. He taught aeronautical engineering at Curtiss-Wright during the 1940s but had a parallel career as a scriptwriter of radio shows, drama critic and newspaper editor. Benedict and Nancy Freedman are joint authors of several well-known novels. They had three children, Johanna, Michael and Deborah.
Michael showed exceptional talents in mathematics as he grew up. However, he also enjoyed painting in an expressionist style and when he entered the University of California at Berkeley in 1968 he still had not decided between mathematics and art. Quite quickly he made a firm decision to study mathematics but, near the end of his first year of study, he decided to apply to do graduate studies at Princeton University. Freedman enjoyed playing Go and he knew that the mathematician Ralph Fox at Princeton was a champion Go player. He read Fox's 1963 text Introduction to Knot Theory and included conjectures of his own in his application to Princeton in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.

After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.

Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.

Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th -century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaré conjecture claims that any closed $n$-manifold which is homotopy equivalent to the $n$-sphere must be the $n$-sphere. When $n = 3$ this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for $n$ at least 5. Freedman proved the conjecture for $n = 4$ in 1982 but the original conjecture remained open until settled by G Perelman who was offered the 2006 Fields medal for his proof.

Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-

Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere $S^{4}$, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.

Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-

After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.

Besides these results about closed simply connected four manifolds, Freedman also proved:

1. Any four manifold properly equivalent to $\mathbb{R}^{4}$ is homeomorphic to $\mathbb{R}^{4}$; a related result holds for $S^{3} \times \mathbb{R}$.
2. There is a nonsmoothable closed four manifold.
3. The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.

Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of $\mathbb{R}^{4}$.

In his reply Freedman thanked his teachers (whom he said included his students) and also gave some fascinating views on mathematics [3]:-

My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.

In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-

In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.

In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.

Freedman continued to hold the Charles Lee Powell Professorship of Mathematics at the University of California at San Diego until 1998 when he left the academic world to take up an appointment with Station Q, a Microsoft research group working on topological quantum computing. In doing so, Freedman became the first Fields Medallist to leave the academic world to work for a company. Freedman became the director of Station Q and to see a little of the topics that he worked on we will look at a few of the titles of the papers he has written over the succeeding ten years. First we note that he gave an invited address to the International Congress of Mathematicians in Berlin in 1998 on Topological views on computational complexity. Then he published papers such as: Quantum computation and the localization of modular functors (2001); Projective plane and planar quantum codes (2001); Poly-locality in quantum computing (2002); Simulation of topological field theories by quantum computers (2002); Topological quantum computation (2003); Approximate Counting and Quantum Computation (2005); Topological quantum computation (2006); Topological quantum computing with only one mobile quasiparticle (2006); Interacting anyons in topological quantum liquids: The golden chain (2008); Measurement-Only Topological Quantum Computation (2008); and Topological Phase in a Quantum Gravity Model (2008). He also published some papers which were not related to quantum computing such as Extension of incompressible surfaces on the boundaries of 3-manifolds (2000), Diameters of Homogeneous Spaces (2003), and Covering a nontaming knot by the unlink (2007).