斯蒂芬·斯梅尔Stephen Smale

斯蒂芬·斯梅尔出生在密歇根州东部的弗林特，这是一个以通用汽车公司所在地而闻名的城镇。从五岁起，他就住在农场里，而他的父亲则在城里为通用汽车公司工作。有八年时间，斯梅尔在离他家农舍大约一英里的小学上学。这所学校只有一个教室，因此老师不得不在困难的情况下教育所有年龄段的孩子。此后，他在高中学习，最喜欢的科目是化学。进入密歇根大学时，他的兴趣已转向物理学，但在物理课程不及格后，他转向了数学。他于1952年获得学士学位，次年获得硕士学位。

斯梅尔继续在安娜堡的密歇根大学攻读博士学位，师从R 拉乌尔·博特，并于1957年凭借学位论文Regular Curves on Riemannian 流形获得博士学位。在论文中，他将哈斯勒·惠特尼于1937年证明的平面曲线结果推广到了$n$维流形上的曲线。斯梅尔在1956-58年间是芝加哥大学的讲师。

1958年，斯梅尔了解到列夫·庞特里亚金关于结构稳定向量场的工作，并开始应用topological方法来研究这些问题。随后，他在1958至1960年间，凭借国家科学基金会博士后奖学金，在普林斯顿的高等爱德华·斯图迪研究所工作。奖学金的最后六个月，他在里约热内卢的纯粹与应用数学研究所度过。在那里，他收到了诺曼·莱文森的一封信，这促使他研究混沌现象；我们将在下面描述这些思想。

1960年，斯梅尔被任命为乔治·伯克利加利福尼亚大学的数学副教授，次年转任哥伦比亚大学教授。1964年，他回到伯克利加利福尼亚大学担任教授，其职业生涯的大部分时间都在此度过。他于1995年从伯克利退休，并担任香港城市大学教授。

1966年，斯梅尔在莫斯科国际数学家大会上被授予菲尔兹奖。导致这一奖项的工作由René 勒内·托姆描述，见[9]。斯梅尔令人印象深刻的成果之一是他关于广义儒勒·昂利·庞加莱猜想的工作。

此处一段未译出，以下为英文原文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 $n$-dimensional 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.
(Michael 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.)

斯梅尔做出重大贡献的另一个领域是莫尔斯理论，他将其应用于多重积分问题。事实上，斯梅尔利用莫尔斯理论攻克了广义儒勒·昂利·庞加莱猜想。

斯梅尔的另一个发现与奇怪吸引子有关。经典力学中的吸引子是一种描述动力系统行为的几何方式。有三种经典吸引子：表征稳态系统的点、表征周期系统的闭合环，以及结合多个循环的环面。斯梅尔发现了导致混沌动力系统的奇怪吸引子。奇怪吸引子在所有放大尺度上都有精细结构，是最早被研究的分数维图形之一。

斯梅尔的职业生涯在20世纪60年代末改变了方向，见[2]：-

到六十年代末，斯梅尔已转向应用领域。他用动力系统对物理过程进行建模，开辟了新的探究方向。$n$体问题和电路理论是斯梅尔用动力系统语言表述的应用之一。在七十年代的大部分时间里，Steve专注于经济学，将拓扑学和动力学注入一般经济均衡的研究。在确立了均衡的性质之后，斯梅尔开始思考计算均衡的算法。传统上算法的收敛理论是局部的，而斯梅尔为这些问题引入了全局视角。算法是否相当可靠，以及预期需要多少次迭代？……

斯梅尔最近的工作是理论计算机科学。他与合作者L Blum和M Shub一起，开发了一种计算模型，该模型既包括艾伦·图灵机方法，也包括数值分析的数值方法。

斯梅尔因其工作获得了许多荣誉。除了上述约翰·查尔斯·菲尔兹奖章外，他还于1966年被美国数学会授予奥斯瓦尔德·维布伦几何学奖：-

……因他对微分拓扑各个方面的贡献。

1996年，斯梅尔获得了国家科学奖章（[2]），因为：-

……四十年来在基础研究问题上的开创性工作，这些工作导致了纯数学与应用数学的重大进展。

斯梅尔的贡献在[2]中得到了很好的总结：-

在整个职业生涯中，斯梅尔处理数学问题时，具有向他人学习的学识、不受传统智慧约束的胆识，以及采用新方法和构建原创框架的能力与远见。事后看来，斯梅尔的发展显得如此自然，然而却没有其他人想到它。

斯梅尔获得了以下大学授予的荣誉学位：华威大学（1974年）、安大略省金斯顿的皇后大学（1987年）、密歇根大学（1996年）、巴黎皮埃尔和玛丽·居里大学（1997年）、香港城市大学（1997年）、罗斯托夫国立大学（1999年）和热那亚大学（2004年）。他还被选为里约热内卢纯粹与应用数学研究所（1990年）、都柏林三一数学学会（1991年）、Moscow Mathematical Society（1997年）和伦敦数学会（1998年）的荣誉会员。

除上述奖项外，斯梅尔还于1988年因论文On the Efficiency of Algorithms in Analysis获得美国数学协会颁发的肖维勒奖。次年，他获得工业与应用数学学会颁发的冯·诺伊曼奖，该机构还在2005年授予他尤尔根·莫泽尔奖。随后在2007年，他荣获沃尔夫奖：-

……因其开创性贡献在微分拓扑学、动力系统、数学经济学以及数学其他分支的形成中发挥了根本性作用。

最后我们注意到，斯梅尔继续担任香港城市大学杰出大学教授直至2001年。他于2002年被任命为芝加哥丰田技术学院教授。随后在2009年，他再次被任命为香港城市大学杰出大学教授。

Stephen Smale was born in Flint in east Michigan, a town famed as the site of General Motors, the automobile company. From the age of five he lived on a farm while his father worked in the city for General Motors. For eight years Stephen attended elementary school about a mile from his farmhouse. This school only had a single classroom, so has teacher had to cope with educating children of all ages under difficult circumstances. Following this, he studied at high school where his favourite subject was chemistry. His interests had moved to physics by the time he entered the University of Michigan, but after failing a physics course he turned to mathematics. He was awarded a BS in 1952 and an MS the following year.

Smale continued to work for his doctorate at the University of Michigan, Ann Arbor under R Bott's supervision and he was awarded his Ph D in 1957 for the thesis Regular Curves on Riemannian Manifolds. In his thesis he generalised results proved by Whitney in 1937 for curves in the plane to curves on an $n$-dimensional manifold. Smale was an instructor at the University of Chicago in 1956-58.

In 1958 Smale learnt about Pontryagin's work on structurally stable vector fields and he began to apply topological methods to study the these problems. He the spent the years 1958-60 at the Institute for Advanced Study at Princeton on a National Science Foundation Postdoctoral Fellowship. The last six months of this fellowship he spent at the Instituto de Mathematica Pura e Aplicada in Rio de Janeiro. Here he received a letter from Levinson which led him to study chaotic phenomena; we describe these ideas below.

In 1960 Smale was appointed an associate professor of mathematics at the University of California at Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at the University of California at Berkeley where he has spent the main part of his career. He retired from Berkeley in 1995 and took up a post as professor at the City University of Hong Kong.

Smale was awarded a Fields Medal at the International Congress at Moscow in 1966. The work which led to this award was described by René Thom, see [9]. One of Smale's impressive results was his work on the generalised 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 $n$-dimensional 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.
(Michael 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.)

Another area in which Smale has made a very substantial contribution is in Morse theory which he has applied to multiple integral problems. In fact Smale attacked the generalised Poincaré conjecture using Morse theory.

Another discovery of Smale's related to strange attractors. An attractor in classical mechanics is a geometrical way of describing the behaviour of a dynamical system. There are three classical attractors, a point which characterises a steady state system, a closed loop which characterises a periodic system, and a torus which combines several cycles. Smale discovered strange attractors which lead to chaotic dynamical systems. Strange attractors have detailed structure on all scales of magnification and were one of the early fractals to be studied.

Smale's career changed direction in the late 1960s, see [2]:-

By the late sixties Smale had moved into applications. He modelled physical processes by dynamical systems, opening new lines of inquiry. The $n$-body problem and electric circuit theory were among the applications that Smale framed in the language of dynamical systems. For much of the seventies Steve focused on economics, injecting topology and dynamics into the study of general economic equilibria. Having established the nature of equilibria, Smale began to think algorithms for their computation. While traditional approaches to the convergence theory of algorithms were local, Smale introduced a global perspective to the problems. Was the algorithm reasonably reliable, and how many iterations were to be expected? ...

Smale's recent work has been on theoetical computer science. With co-workers L Blum and M Shub, he has developed a model of computation which includes both the Turing machine approach and the numerical methods of numerical analysis.

Smale has received many honours for his work. In addition to the Fields Medal described above, he was awarded the Veblen Prize for Geometry by the American Mathematical Society in 1966:-

... for his contributions to various aspects of differential topology.

In 1996 Smale received the National Medal of Science ([2]) for:-

... four decades of pioneering work on basic research questions which have led to major advances in pure and applied mathematics.

Smale's contribution is nicely summed up in [2]:-

Throughout his career Smale has approached mathematical problems with the scholarship to learn from others, the audacity to be unconstrained by conventional wisdom, and the power and vision to employ new methods and construct original frameworks. After the fact, a Smale development seems so natural, yet no one else thought of it.

Smale has been awarded honorary degrees by the University of Warwick (1974), Queens University, Kingston, Ontario (1987), the University of Michigan (1996), Université Pierre et Marie Curie, Paris (1997), the City University of Hong Kong (1997), Rostov State University (1999), and the University of Genoa (2004). He has been made an honorary member of the Instituto de Matematica Pura e Aplicada, Rio de Janeiro (1990), Trinity Mathematical Society, Dublin (1991), Moscow Mathematical Society (1997) and the London Mathematical Society (1998).

In addition to the prizes mentioned above, Smale has been awarded the Chauvenet Prize by the Mathematical Association of America in 1988 for his paper On the Efficiency of Algorithms in Analysis. In the following year he was awarded the Von Neumann Award by the Society for Industrial and Applied Mathematics who also presented him with their Jürgen Moser Prize in 2005. Then in 2007 he was honoured with the award of the Wolf Prize:-

... for his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics, and other subjects in mathematics.

Finally we note that Smale continued as Distinguished University Professor at the City University of Hong Kong until 2001. He was appointed Professor at the Toyota Technological Institute at Chicago in 2002. Then in 2009 he was again appointed Distinguished University Professor at the City University of Hong Kong.