10/20/2004

Alexandre Kirillov
University of Pennsylvania

Analysis and geometry of fractals

I discuss several interesting phenomena (geometric, algebraic, analytic and arithmetic) which are observed in the study of fractal sets and functions on them. Only two examples will be considered: Sierpinski and Apollonian
gaskets.
Main topics:
1. Definition and analytic structure of harmonic functions on the Sierpinski gasket.
2. Arithmetic properties of values of harmonic functions on the Sierpinski gasket.
3. Definition of Apollonian gaskets and geometry of circles on spheres. Generalized Descartes theorem.
4. Arithmetic properties of Farey series and curvatures of circles in Apollonian gaskets.
5. The Minkowski's "question function".