Consider the n-cube graph in R^n, with vertices {0,1}^n and edges connecting vertices with Hamming distance 1.
How many hyperplanes are required in order to dissect all edges?
This problem has been open since the 70s. We will discuss this and related problems.

Puzzle: Show that n hyperplanes are sufficient, while sqrt(n) are not enough.

In this talk we will discuss some results about the regularity of maximal operators on Sobolev and BV spaces. We will also discuss many related open problems.

Masani and Wiener asked to characterize the regularity of vector stationary stochastic processes. The question easily translates to a harmonic analysis question: for what matrix weights the Hilbert transform is bounded with respect to this weight? We solved this problem with Sergei Treil in 1996 introducing the matrix A_2 condition.

But what is the sharp estimate of the Hilbert transform in terms of matrix A_2 norm? This is still unknown in a striking difference with scalar case.

Convex body valued operators helped to get the estimate via norm raised to the power 3/2. But shouldn't it be power 1? 

We construct an example of a rather natural operator for which the estimate in scalar and vector case is indeed different. But it is not the Hilbert transform.

Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santaló type inequality (we call it $j$-Santal\'{o} conjecture)  for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball. 
Our results also strengthen one of the main results of Kolesnikov-Werner, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be interpreted as generalizations of the classical Blaschke-Santaló inequality.
Related, we discuss an analogue of a conjecture due to K. Ball in the multi-entry setting and establish a connection to the $j$-Santaló conjecture.

In this talk, I will present several geometric and analytic characterizations of purely unrectifiable quasicircles. These necessary and sufficient conditions are expressed in terms of various notions such as Dirichlet algebras, harmonic measure, analytic capacity and continuous analytic capacity. As an application, I will explain how to construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion. This answers a question raised by Paul Gauthier.