Instructor: Dr. Márton ELEKES
K. Falconer: Fractals. A very short introduction. Oxford University Press, 2013.
P. Mattila: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge University Press, 1995.
P. R. Halmos: Measure Theory. D. Van Nostrand, 1950.
J. C. Oxtoby: Measure and category. A survey of the analogies between topological and measure spaces. Springer, 1980.
W. Rudin: Functional analysis. McGraw-Hill, 1973.
Prerequisits: introductory courses to measure theory, group theory and topology
Fractal geometry: self-similar sets, Hausdorff measures, Hausdorff dimension, box dimension.
Haar measure: existence, uniqueness, modular function, Pontryagin duality and the structure of locally compact groups.
Lipschitz functions: Theorems of Kirszbraun and Rademacher, the cases of Euclidean, Hilbert and Banach spaces.
Genericity and prevalence: Baire category, Christensen's notion of Haar null sets in non-locally compact groups, constructions of exotic objects.