Mathematical Sciences: Measure and Dimension, Scaling, Random Objects and Selections

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Professor Mauldin will conduct research in various aspects of the measure theory of fractal objects. One major theme centers on Hausdorff measure and its variants. Topics include (i) further development of deterministic and random recursions yielding objects in n-space and determining their Hausdorff and packing dimension functions, (ii) finding the exact Hausdorff dimension of the graphs of general Hardy-Weierstrass functions, and (iii) characterizing Hausdorff measures which scale. A second theme centers on the production of random maps and finding their properties. Most smooth objects in space have an integer dimension. For example a solid object has dimension three and a surface dimension two. Hausdorff dimension and measure are capable of describing fractal objects which have non-integer dimension.