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What are some examples of cutting-edge research involving fractals or self-similar structures?

Who's actively contributing high-quality research in this field?

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    Your question seems a little ambiguous. Self-similarity is a very general phenomena and it happens in many places, not just with objects that are "geometric". You can have self-similarity of algebras and a variety of other kinds of objects. In that sense self-similarity isn't so much of an object of study in its own right, just a general phenomenon. The theorems about the Mandelbrot set that I know of are due to people like Fatou, Julia, Douady and Hubbard, which in some sense are before and after Mandelbrot, respectively. Could you be a little more clear about which field "this field" is?2010-09-26
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    When I look at what Mandelbrot has done it seems his work is mostly recognised as physics, not mathematics. In that he popularized the notion of fractals as being relevant to physics, and physical sciences.2010-09-26

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