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学术生涯

曼德博于2007年演讲时拍摄

1951年后，曼德博的研究范围不仅限于数学理论，更扩展到了信息理论、经济学、流体力学等应用领域。他坚信，厚尾现象和自相似性结构作为两大关键词，能够解决这些领域中的很多问题。 From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory, economics, and fluid dynamics. He became convinced that two key themes, fat tails and self-similar structure, ran through a multitude of problems encountered in those fields.

曼德博发现，金融市场中的价格分布并不服从正态分布，而是服从理论上方差无穷大的稳定分布：例如，相对于 $\alpha =2$ （即为正态分布），棉花价格更接近 $\alpha =1.7$ 的稳定分布。“稳定”分布的性质之一就是n个服从稳定分布的独立同分布的随机变量之和服从尺度参数更大的稳定分布。

Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter.

曼德博对宇宙学领域亦有贡献。1974年，他从分形学角度提出了奧伯斯佯谬（夜黑佯谬）的一种新的充分但不必要的解释。他认为，假定宇宙中恒星的分布是分形集（如康托尘埃），那么奥伯斯佯谬的解释就不必依赖大爆炸理论。他的模型没有把大爆炸理论完全排除在外，但即使大爆炸没有发生过，仍然可以解释夜黑现象。

Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor Dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred. .

1975年，曼德博提出了“分形”一词用于描述这类结构，并在著作《分形学：形态，概率和维度》（原书《Les objets fractals, forme, hasard et dimension》，1975年出版；英译本《Fractals: Form, Chance and Dimension》，1977年出版）阐述了他的观点，同时发展了捷克地理学家、人口学家、统计学家Jaromír Korčák在1938年发表的论文《两种类型的统计分布》（原文《Deux types fondamentaux de distribution statistique》，英译版《Two Basic Types of Statistical Distribution》）中的思想。

In 1975, Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer, demographer and statistician Jaromír Korčák.

曼德博集合及其轨道周期性。

1979年，在哈佛大学作为访问学者的期间，曼德博开始研究分形集之一——在复平面上一定变换下具有不变性的朱利亚集合。在加斯顿·朱利亚和皮埃尔·法图学术成果的基础上，曼德博利用公式 $f_{c}(z)=z^{2}+c$ 反复迭代，在计算机上作出了朱利亚集合的图形。在研究朱利亚集合的拓扑结构是怎样依赖于复参数$\mu$的同时，他还

While on secondment as Visiting Professor of Mathematics at Harvard University in 1979, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets of the formula z² − μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set fractal that is now named after him. (Note that the Mandelbrot set is now usually defined in terms of the formula z² + c, so Mandelbrot's early plots in terms of the earlier parameter μ are left–right mirror images of more recent plots in terms of the parameter c.)

In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature. This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".

Mandelbrot speaking about the Mandelbrot set, during his acceptance speech for the Légion d'honneur in 2006

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003, and the Einstein Lectureship of the American Mathematical Society in 2006.

The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Knight in the French Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was promoted to Officer of the Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.