Remez-type inequalities bound the suprema of low-degree polynomials over some domain K by their suprema over a subset S of K. Existing multi-dimensional Remez inequalities bear constants with strong dependence on dimension. In this talk we will prove a dimension-free Remez-type estimate when K is the polydisc D^n and S is from a certain class of discrete subsets. As a direct consequence we also obtain a Bohnenblust-Hille-type inequality for products of cyclic groups, which in turn has consequences for learning algorithms. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.

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.