Vilma Orgoványi (BME)
Title: 
Orthogonal projections of the random Menger sponge

Abstract: 
To model turbulence, Mandelbrot introduced a family of statistically self-similar random sets E which is now called fractal percolation or Mandelbrot percolation. This is a two-parameter (M, p) family of random sets in R^d, where M>1 is an integer and 0<p<1 is a probability. The inductive construction of E is as follows. The closed unit cube of R^d is divided into M^d congruent cubes. Each of them is retained with probability p and discarded with probability 1-p.

In the retained cubes, we repeat this division and retaining/discarding process independently of everything ad infinitum or until there are no retained cubes left. The random set E that remains after infinitely many steps (formally: the intersection of the unions of retained cubes on all construction stages) is the fractal percolation set.


Using an analogous method it is possible to define more general random fractals for example the random Menger sponge. In this talk, I will study the orthogonal projections of the random Menger sponge to straight lines from the point of the positivity of the Lebesgue measure, and the existence of interior points.
This is based on joint work with Károly Simon.