My research may be classified into three main areas: (i) development and application of fractal geometric ideas to hydrology and geophysics, (ii) study of an unforeseen route to the Gaussian distribution via iterations of simple maps, and (iii) investigation of linkages between natural and man-made complexity with applications in the field of science and faith.
With my students and collaborators we have developed a new deterministic approach aimed at addressing the ever elusive problem of geophysical (hydrologic) complexity. Reminiscent of the Platonic notion of reality being a "shadow,” the ideas rely on studying the many interesting projection patterns obtained while using multifractal measures to illuminate fractal interpolating functions.
We have established that such notions give rise to a host of “natural” patterns over one, two and three dimensions, that may be used to model a variety of complex geophysical sets, including rainfall time series, pollution spatial patterns, and both chaotic and "random" time series.
Experimenting with these same ideas, we discovered that arbitrary illumination of space-filling fractal interpolating functions always result in Gaussian distributions, over one or higher dimensions. This surprisingly universal result linking the famous bell with fractals provides, in addition, an unforeseen bridge between disorder and order, one that is counterintuitive, for the notions relate turbulence with diffusion, and dissipation with conduction.
While attempting to prove the validity of the Gaussian result over two dimensions, we stumbled upon the stunning presence of a host of "Treasures inside the bell." These are beautiful and exotic mathematical designs that decompose the circular Gaussian distribution into crystalline patterns of arbitrary symmetries, which surprisingly include the geometric structure of nature’s ice crystals and also biochemical rosettes, comprising even life’s own DNA.
Inspired by our work on the Gaussian distribution and other recent universal results pertaining to natural complexity, I have been studying the applicability of such laws to understand man-made complexity, hoping to establish from them useful lessons for inner and world peace.