I had the privilege to work as a post-doc at the University of Edinburgh with Dr. Ben Hambly in 1996-98, after I finished my PhD thesis. Hambly was the main opponent at my thesis defense, and found the directions of my research interesting. The work we did in Edinburgh was to a large extent an extension of my doctoral thesis research.
My work
was on Brownian motion on
fractals. We studied the properties of Brownian motion (BM) because BM
encodes information about the so called Laplace operator, and hence about
very many physical properties of the fractal. We study fractals because we
see them as approximate models for real physical media - especially for
porous or "disordered" media. So the study of Brownian motion on
fractals is one gateway to understanding the physical properties of porous
physical objects.
Before our studies, mathematicians had studied BM on fractals or classes of fractals, but they had not compared the BM on different fractals. If we want to apply this kind of mathematics to modelling real phenomena, such a comparison is unavoidable: Say we want model a physical object by a fractal. We will never find a precise match, and so we might have two fractal approximations to our object. Two "almost similar" fractals. What if the two models give wildly different descriptions of the physical processes? If our theory of modelling is good, the two fractals should yield almost the same results in some way. If the results may diverge, our studies will not be of much value in modelling real objects.
The bulk of my work has been centered on showing when "a small change" in the fractal really is also a small change from the point of view of physical processes. The GIF movie displayed on this page shows what seems like a loop in fractal space. Visually, we see small, continuous changes of the fractal. Our immediate guess would be that the corresponding physical properties would undergo small, continuous changes along with the fractal. This does, however, turn out not to be the case. There is one point in the loop where a change occurs that leads to a leap in certain physical measurements. The reader is challenged to guess where this is for herself, and then click on the picture for the solution.
A brief synopsis of my results: I have looked at a set of certain finitely ramified fractals, and made this into an abstract space by defining a metric on this set. This metric is defined so that if a sequence of fractals kn converge to a fractal k in this metric, then the law of the diffusions on fractals kn converge to the law of the diffusion on fractal k. The comparisons of fractals and diffusions on them is made via maps called fractomorphisms - maps that preserve fractal structure. More technically, I define a distance between (the law of) processes on a fractal, and then declare the distance between BMn on kn and BMm on km to be the least possible distance on km between BMm and the map of BMn via any fractomorphism from kn to km.
In connection with my work, I have introduced some concepts and tools that I believe will ease work of some mathematical tasks. They will of course not ease the mathematician's work, as a mathematician, once he has simplified something, will go for something else or an extension which is more difficult. The two most useful concepts, I believe, are those of multigraphs and of fractomorphism (descriptions to be implemented later). I have also devised a class of fractals known as Simple Fractal spaces which are a useful experimenting ground for researchers in this field who want to test out new ideas. With the aid of fractomorphisms, I have also defined a metric on the "space of spaces" of SF fractals, so that convergence in space also means convergence in the basic diffusion process - Brownian motion. That is the kind of problem I have been working on in Edinburgh.
Conferences: Look at the picture of NSA '96 participants. I'm the guy in the very visible t-shirt. NSA = NonStandard Analysis - the type of mathematical analysis where we are using the techniques associated with infinitesimals in preference to the techniques based on the limit concept.