Abstract:
\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If $\dim_{\rm H}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\geq 2$ one-dimensional fractal percolations.

Abstract:
Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.

Abstract:
In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals.

Abstract:
We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.

Abstract:
We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu's result \cite{FeHu09} for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures \cite{HoSh12} to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, $C^1$-images and distance sets.

Abstract:
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.

Abstract:
Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given integer m between 2 and n - 1 (respectively, between 3 and n - 1), the orthogonal projections of the sets on every m-dimensional plane are homothetic, where homothety ratio and its sign may depend on the projection plane. The proof uses a refined version of Straszewicz's theorem on exposed points of compact convex sets.

Abstract:
Intersection of a random fractal or self-affine set with a linear manifold or another fractal set is studied, assuming that one of the sets is in a translational motion with respect to the other. It is shown that the mass of such an intersection is a self-affine function of the relative position of the two sets. The corresponding Hurst exponent h is a function of the scaling exponents of the intersecting sets. A generic expression for h is provided, and its proof is offered for two cases --- intersection of a self-affine curve with a line, and of two fractal sets. The analytical results are tested using Monte-Carlo simulations.

Abstract:
The power-law dependence of the angle in the angular projection of galaxy distribution is explained by assuming that in the spherical shells within a small angle the distributions are also fractal. If this local angular fractal is possessed, a fractal structure is angularly-isotropic at each occupied point though inhomogeneous, and is compatible with the present evidence claimed to be of homogeneity for galaxy distribution. Further, it is most likely to be isotropic rather than only angularly-isotropic. Several related issues are discussed.

Abstract:
The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis.