# Geometry and analysis on fractals

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We consider only two fractals: Sierpinski and Apollonian gaskets. The

idea is to show on these two examples how geometry, analysis, algebra and

number theory are tied together in the simplest problems, related to

fractal sets.

We start with definitions, speculate on the general matrix numerical

systems, consider the analytic properties and the p-adic behavior of

harmonic functions, analyse the spectrum of the Laplace operator on the

Sierpinski gasket. Then we describe the geometry, group-theoretic

structure and arithmetic properties of the Apollonian gasket.

The final idea is to draw a parallel between the two fractals - an

unfinished program.