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Mandelbrot, BenoitYale University
Mandelbrot is widely recognized for introducing the concept of fractals into fields of physics that defied description by conventional concepts and for his impact on these fields. He demonstrated that the concept of non-integral dimension is the key to describing a vast array of physical phenomena and objects with no characteristic length scale. His creative thinking was essential for physicists working in diverse fields to capture the essence of previously intractable problems and to unveil order and simplicity in systems with a seemingly high degree of disorder irregularity, and complexity.
Elected to NAS: 1987Scientific Discipline: Applied Physical Sciences Membership Type: Member
Research Interests:аI conceived and developed a new geometry of nature and implemented it in different fields. Fractal geometry builds on shapes having identical structure at all scales. Several were known, but called "pathological." I discovered that this "pathology" rules parts of nature and also the stock market and that it can be very beautiful. When the singularities and/or boundaries of partial differential equations are movable, I argued that they eventually become fractal. Adding that attractors are typically fractal, fractality is often explainable by "completing" the usual smooth dynamics. I introduced the multifractals, negative dimensions, pulse processes, squigs, fractal lacunarity, and R/S statistics. Added to fractals, those new tools helped describe and understand the galaxies' distribution, turbulence, percolation, 1/f noises, fractal aggregation, etc. Graphics led me to mathematical conjectures (the main one remains unproven) concerning quadratic dynamics (the Mandelbrot set) and the boundary of planar Brownian motion (namely, its dimension is 4/3). Two basic features I identified in finance are prices' discontinuity and concentration. I accounted for them by introducing the powerful fractal tools of scaling, renormalization, and long-range dependence (before their use in physics) and by proposing models that fit well.
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