Benoît B. Mandelbrot (2007)

Benoît B. Mandelbrot (born November 20, 1924 in Warsaw , † October 14, 2010 in Cambridge , Massachusetts ) was a French - American mathematician.

Mandelbrot made contributions to a wide range of mathematical problems, including theoretical physics , financial mathematics, and chaos research . But he was best known as the father of fractal geometry . He described the amount of Mandelbrot and coined the term "fractal". Mandelbrot himself contributed greatly to popularizing his work by writing books and giving lectures that were intended for the general public.

Mandelbrot spent most of his career at IBM's Thomas J. Watson Research Center , where he was an IBM Fellow . He later became Sterling Professor of Mathematics ( Mathematical Sciences ) at Yale University . He was also a research fellow at the Pacific Northwest National Laboratory , the University of Lille I , the Institute for Advanced Study and the Center national de la recherche scientifique . Mandelbrot lived in the United States until his death.

## Early life

The amount named after Mandelbrot with a colored area

Mandelbrot was born in Poland to a Lithuanian-Jewish family with an academic tradition. His mother was a doctor, his father a clothes dealer. As a boy, Mandelbrot was introduced to mathematics by two uncles, one of whom, Szolem Mandelbrojt , taught mathematics at the Collège de France . In 1936 the family moved to Paris to avoid the looming threat from the National Socialists .

Mandelbrot attended the Lycée Rolin in Paris until the outbreak of World War II . He earned a reputation for mathematical talent through his ability to visualize tasks as geometric problems. In a national test, he was the only student in France to solve a math problem. According to his own statement, he did not even try to calculate the complicated integral , but instead recognized that the task was based on a circular formula and transformed the coordinates in order to use the circle in the solution. His family fled from the German occupation to Vichy France , to Tulle , where the Rabbi von Brive-la-Gaillarde supported him in his schooling. From 1945 to 1947 he studied engineering at the École polytechnique with Gaston Julia and Paul Lévy . He then completed a degree in aeronautics at the California Institute of Technology , which he completed in 1949 with a master's degree.

After completing his studies, Mandelbrot returned to France and received his doctorate in mathematics from the University of Paris in 1952 . From 1949 to 1957 he was a research assistant at the Center national de la recherche scientifique . During this time Mandelbrot spent a year at the Institute for Advanced Study in Princeton, New Jersey , where he was assisted by John von Neumann . In 1955 Mandelbrot married Aliette Kagan, moved with her to Geneva and then back to France. After a year at Université Lille Nord de France , Mandelbrot joined the research department at the Thomas J. Watson Research Center at IBM in 1958 , where he was made an IBM Fellow in 1974, an award which gave him extensive freedom in his research.

## Scientific career

Mandelbrot at a speech in 2007

From 1951, Mandelbrot published works on problems in mathematics, but also on problems in applied areas such as information theory , economics and fluid mechanics . He was increasingly convinced that a large number of problems in these areas were determined by two central themes, namely “fat tail” probability distributions and self-similar structures.

Mandelbrot found that the price fluctuations of the financial markets cannot be described by a normal distribution , but by a Lévy distribution , which theoretically has an infinite variance . For example, he showed that cotton prices since 1816 have followed a Lévy distribution with the parameter , while they would correspond to a Gaussian distribution (see also alpha-stable distributions ). He also provided a possible explanation for the Equity Premium Puzzle . ${\ displaystyle \ alpha = 1.7}$${\ displaystyle \ alpha = 2}$

Mandelbrot also applied these ideas in the field of cosmology . In 1974 he proposed a new explanation for Olbers' paradox of the dark night sky. He showed that the paradox can be avoided even without recourse to the Big Bang theory, if one assumes a fractal distribution of the stars in the universe, in analogy to the so-called Cantor dust .

In 1975 Mandelbrot coined the term fractal to describe such structures. He published these ideas in the book Les objets fractals, forme, hasard et dimension (1975; an English translation of Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed ideas of the Czech geographer, demographer and statistician Jaromír Korčák (1895-1989), which he had published in the article Deux types fondamentaux de distribution statistique (1938; German two basic types of statistical distribution ).

During his tenure as visiting professor of mathematics at Harvard University in 1979, Mandelbrot began studying the fractal Julia sets , which are invariant to certain transformations in the complex plane and which were previously studied by Gaston Julia and Pierre Fatou . These quantities are generated by the iterative formula . Mandelbrot used computer plots of this set to investigate their topology as a function of the complex parameter . In doing so, he discovered the amount of Mandelbrot named after him. ${\ displaystyle z ^ {2} + c}$${\ displaystyle c}$

Mandelbrot giving his acceptance speech after being promoted to officer of the Legion of Honor, 2006

In 1982 Mandelbrot expanded his ideas and published them in his most famous book The Fractal Geometry of Nature (the German translation was published in 1987 under the title The fractal geometry of nature ). This influential book brought fractals to a wider audience and also silenced many of the critics who had previously dismissed fractals as a programming artifact .

Mandelbrot left IBM in 1987 after 35 years with the company after IBM decided to close its basic research department. He then worked in the mathematics department at Yale University , where he took up his first permanent professorship in 1999 at the age of 75. When he retired in 2005, he was Sterling Professor of Mathematics. His last position was in 2005 as a Battelle Fellow at the Pacific Northwest National Laboratory . In the same year, the Deutsche Bundesbank organized a celebratory colloquium on the occasion of his 80th birthday on the subject of Heavy tails and stable Paretian distributions in finance and macroeconomics , in order to recognize his contributions to a better understanding of financial markets and financial market stability.

His most recent publication on fractal mathematical structures in the financial markets of fractals and finance won the Financial Times Germany's business book award .

## Fractals and roughness in nature

Although Mandelbrot coined the term fractal , some of the objects depicted in The Fractal Geometry of Nature were described earlier by mathematicians. Before Mandelbrot, however, they were viewed more as unnatural mathematical oddities. It was Mandelbrot's merit to use fractal geometry for the description of real objects, whose "rough" objects, which could not be described by simple idealizations, eluded scientific investigation until then. He showed that all of these objects have certain properties in common, such as self-similarity , scale invariance and often a non-integer dimension . Examples of natural fractals are the shapes of mountains, coastlines and rivers, branches of plants, blood vessels and alveoli , the distribution of star clusters in galaxies and the paths of Brownian motion . Fractal structures can also be found in quantitative descriptions of human creativity and actions, for example in music , painting and architecture as well as in stock exchange prices . Mandelbrot was therefore of the opinion that fractals are much more amenable to intuitive comprehension than the artificially smoothed idealizations of traditional Euclidean geometry :

“Clouds are not spheres, mountains are not cones, coastlines are not circles. The bark is not smooth - and lightning does not make its way straight either. "

- Mandelbrot : The Fractal Geometry of Nature

Mandelbrot was called a visionary and an independent spirit (English "Maverick"). His generally understandable and passionate writing style and his emphasis on pictorial geometric intuition made his book The Fractal Geometry of Nature in particular accessible to non-scientists. The book sparked wide public interest in fractals and chaos theory .

## death

Mandelbrot died of pancreatic cancer at the age of 85 . On the occasion of his death, the mathematician Heinz-Otto Peitgen described Mandelbrot as one of the most important personalities of the last 50 years for mathematics and its application in natural science. French President Nicolas Sarkozy praised Mandelbrot as a great and original spirit whose work went completely beyond the scientific mainstream. The Economist magazine also pointed to his fame beyond science, calling him the father of fractal geometry.

## Trivia

The name of the German live coding band Benoît and the Mandelbrots refers to the famous mathematician. The group was founded in Karlsruhe in 2009.

## Fonts

• with Richard Hudson: Fractals and Finance: Markets Between Risk, Return, and Ruin. Piper, 2007 (English original: The (mis) behavior of markets: a fractal view of risk, ruin and reward. Basic Books, 2004)
• The fractalist: memoir of a geometer. Pantheon Books, 2012.
• Selecta. several volumes, Springer Verlag, from 1997.
• The fractal geometry of nature. Birkhäuser, 1987, 2001.
• Fractals: Form, Chance and Dimension. Freeman, San Francisco 1977.
• The fractal geometry of nature. Freeman, 1983.
• Nice chaos: my wondrous life. Piper Verlag, Munich 2013, ISBN 978-3-492-96162-2 . (Autobiography)

## literature

• Donald J. Albers, GL Alexanderson: Mathematical People - Profiles and Interviews. Birkhäuser 1985.
• Nigel Lesmoir-Gordon: Clouds are not Spheres. A Portrait of Benoît Mandelbrot, the Founding Father of Fractal Geometry. World Scientific, London 2018, ISBN 978-1-78634-474-8 .

Commons : Benoît Mandelbrot  - album with pictures, videos and audio files

## Individual evidence

1. ^ Jascha Hoffmann: Benoît Mandelbrot, Novel Mathematician, Dies at 85 . In: The New York Times . October 16, 2010 ( online [accessed April 11, 2013] Mandelbrot did not use an accent for his first name. He chose the initial B himself, but it does not appear as a middle name).
2. ^ A b Benoît Mandelbrot, Novel Mathematician, Dies at 85. In: The New York Times . October 16, 2010, accessed October 18, 2010 .
3. Benoît Mandelbrot, Bernard Sapoval, Daniel Zajdenweber: Web of Stories • Benoît Mandelbrot • Family background and early education. Web of Stories , May 1998, accessed October 19, 2010 .
4. a b c Benoît Mandelbrot: A maverick's apprenticeship . In: Imperial College Press (Ed.): The Wolf Prizes for Physics . 2002 ( PDF ).
5. BBC News - 'Fractal' mathematician Benoît Mandelbrot dies aged 85. In: BBC Online . October 17, 2010, accessed October 17, 2010 .
6. ^ Benoit B. Mandelbrot: The Fractalist: Memoir of a Scientific Maverick. Pantheon, 2013. Quoted in: New York Review of Books: He Conceived the Mathematics of Roughness , May 23, 2013.
7. Mandelbrot's curriculum vitae on his personal homepage. ( MS Word ; 976 kB) Retrieved December 16, 2010 .
8. Benoît B. Mandelbrot: Some Potential Problems Arising in the Theory of Axial Turbomachines . Ed .: California Institute of Technology . Pasadena, California 1949 (English, PDF [accessed October 18, 2010] Thesis for the Degree of Aeronautical Engineer).
9. a b Anthony Barcellos: Interview with BB Mandelbrot . In: Birkhäuser (Ed.): Mathematical People . 1984 ( PDF ).
10. Benoît Mandelbrot: The Variation of Certain Speculative Prices . IBM Research Report NC-87, 1962 ( PDF ).
11. Fractals and Finance: Markets Between Risk, Return, and Ruin. ISBN 3-492-04632-0 , p. 315ff.
12. Benoît Mandelbrot: Fractals: Form, Chance and Dimension. WH Freeman and Co, 1977, ISBN 0-7167-0473-0 .
13. Jaromír Korčák: Two types fondamentaux de distribution statistique. In: Prague, Comité d'organisation, Bull. De l'Institute International de Statistique. vol. 3, 1938, pp. 295-299.
14. ^ Benoît B. Mandelbrot: The Fractal Geometry of Nature . WH Freeman Company, New York, NY 1977 (English).
15. Benoît B. Mandelbrot: The fractal geometry of nature . Birkhäuser, Basel / Boston 1987, ISBN 3-7643-1771-X .
16. Benoît Mandelbrot, Bernard Sapoval, Daniel Zajdenweber: Web of Stories • Benoît Mandelbrot • IBM: background and policies. Web of Stories , May 1998, accessed October 17, 2010 .
17. ^ Edward Tenner: Benoît Mandelbrot the Maverick, 1924-2010. In: The Atlantic . October 16, 2010, accessed October 16, 2010 .
18. PNNL press release: Mandelbrot joins Pacific Northwest National Laboratory. Pnl.gov, February 16, 2006, accessed October 17, 2010 .
19. http://www2.cirano.qc.ca/~dufourj/Web_Site/StabDistConf_2005_11_Bundesbank_Frankfurt.pdf (accessed on May 21, 2016)
20. Fractals and Finance: Markets Between Risk, Return, and Ruin. In: ephorie.de - the management portal. Retrieved October 18, 2010 (Book Review, ISBN 3-492-04632-0 .).
21. On fractals and mathematical art
22. ^ Robert L. Devaney: "Mandelbrot's Vision for Mathematics" in Proceedings of Symposia in Pure Mathematics . Volume 72.1. (PDF; 231 kB) American Mathematical Society, 2004, accessed January 5, 2007 .
23. ^ Bill Jersey: A Radical Mind. In: Hunting the Hidden Dimension. NOVA / PBS, April 24, 2005, accessed August 20, 2009 .
24. ^ Jascha Hoffman: Benoit Mandelbrot, Mathematician, Dies at 85. In: The New York Times. October 16, 2010, accessed October 16, 2010 .
25. ^ Sarkozy rend hommage à Mandelbrot. In: Le Figaro . Retrieved October 17, 2010 (French).
26. Benoît Mandelbrot's obituary (The Economist)
27. Friedemann Dupelius: Dissonance | Show us Your Screens - Benoît and the Mandelbrots: the Live Coding Band from Karlsruhe. Retrieved March 13, 2019 .
28. ↑  Légion d'honneur '' announcement of promotion of Mandelbrot to  officier ''. Legifrance.gouv.fr, accessed October 17, 2010 (French).
29. Member History: Benoît B. Mandelbrot. American Philosophical Society, accessed October 29, 2018 (annotated English).
30. Six granted honorary degrees, Society of Scholars inductees recognized. Gazette.jhu.edu, June 7, 2010, accessed October 17, 2010 .