## Benoît Mandelbrot,

Novel Mathematician, Dies at 85

by Jascha Hoffman, New York Times

Benoît B. Mandelbrot, a maverick mathematician who developed the field of fractal geometry and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. The cause was pancreatic cancer, his wife, Aliette, said. He had lived in Cambridge.

Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.

“Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,” said David Mumford, a professor of mathematics at Brown University. “He was one of the primary people who realized these were legitimate objects of study.”

In a seminal book, “The Fractal Geometry of Nature,” published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as “monstrous” and “pathological.” Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now “be approached in rigorous and vigorous quantitative fashion.”

For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate…

When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.

“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”

Also

The Misbehavior of Markets,

A Fractal View of Risk, Ruin, and Reward

by Benoit Mandelbrot and Richard L. Hudson, Yale.edu, getAbstract

“Fractals and Multifractals

A fractal is a pattern that repeats itself in larger or smaller scale. Consider a fern frond,

for example, or a coastline, or a stock market price chart. A fern leaf looks like a fern

frond. A hundred yards of coastline looks like a hundred miles of coastline. A chart

of stock prices for one day looks like a chart of stock prices for ten years. Fractal

mathematics offers the promise of devising:

• Better tools for investment analysis — Individual stocks may manifest unique fractal

patterns in their price charts — as distinctive as fi ngerprints.

• Superior portfolios — Portfolios can be developed that are better than those derived

through modern portfolio theory. As noted above, conventional thinking about

fi nance assumes that stock prices incorporate all relevant information. It assumes

that prices move randomly. It assumes other things that are clearly at variance with

facts. A new approach would involve, for example, much more extensive stress tests

over a much wider range of possible outcomes.

• Better option valuation models — The “holes” in Black-Scholes, the most widely

accepted option-pricing model, have been the subject of much debate. The holes have

been patched repeatedly, but the patches are wearing thin.

• Risk management models — Assumptions of randomness, normal probability distributions

and so forth underlie many of the risk management models used by international

bankers and regulators. Extensive reliance upon such inadequate models

actually makes the fi nancial world a much more dangerous place.

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