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.