Istvan Kolossvary (TU Budapest):
Fractals in dimension theory and complex networks
(Department defence of the PhD dissertation)
Supervisor: Karoly Simon
Abstract: The main aim of the Thesis is to demonstrate the diverse applicability of fractals in different areas of mathematics. Namely,
1. widen the class of planar self-affine carpets for which we can calculate the different dimensions especially in the presence of overlapping cylinders,
2. perform multifractal analysis for the pointwise H\"older exponent of a family of continuous parameterized fractal curves in $\R^d$ including deRham's curve,
3. show how hierarchical structure can be used to determine the asymptotic growth of the distance between two vertices and the diameter of a random graph model, which can be derived from the Apollonian circle packing problem.
I will present the results in an informal way, illustrated with plenty of examples, and some hints about the heuristics of the proofs.