Geometry of fractals and vector calculus on fractals

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Most people have encountered beautiful pictorial depictions of fractals at one point or another. However, it may come as a surprise to many that fractals play a useful role in scientific areas such as data compression, analyzing stream networks and modelling other physical phenomena. The defining characteristic of a fractal is its self-similarity. That is, small bits of a fractal bear a striking resemblance to the entire fractal. Many natural phenomena have this same type of scaling behaviour, but usually only as an approximation. However, it is often a good enough approximation to be descriptively or predictively useful and so fractals have found their way into most areas of science.** This research project has two main themes, both involving fractals, with the respective goals of a better understanding of the geometry of fractals and building new calculus tools to use when applying fractal-based models. In the first theme a very fine analysis of the geometry of Cantor sets will be undertaken. This is a continuing program to obtain a complete characterization of the Hausdorff and packing measures and dimensions of very general Cantor sets, in particular subsets of the real line. For linear Cantor sets, the approach is to relate the geometry to the asymptotics of the "gaps". In the second theme the new calculus tools will be fractal analogues of the tools from classical calculus such as various types of derivatives and integrals. In this research program, the approach will be to use iterated function systems (IFS) as a constructive tool. In previous work IFS have proven to be useful in describing and constructing fractals in many different contexts. The method will be to adapt the IFS framework to the desired calculus construction, starting with curvatures for fractal curves and differential forms for fractal surfaces and then higher-dimensional analogues of these.** An important outcome of the project will be practical and efficient algorithms to be used in computations involving calculus on fractals. These algorithms will make it possible for scientists to use fractal models more efficiently. This is a primary reason for using IFS methods, as they naturally lead to computational algorithms.****