The classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies
$\|p(T)\| \leq \sup _{|z| \leq 1}|p(z)|$
Whereas Andô extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.[-]

The matrix $A_2$ condition on the matrix weight $W$$$[W]_{A_2}:=\sup _I\left\|\langle W\rangle_I^{1 / 2}\left\langle W^{-1}\right\rangle_I^{1 / 2}\right\|^2<\infty$$where supremum is taken over all intervals $I \subset \mathbb{R}$, and$$\langle W\rangle_I:=|I|^{-1} \int_I W(s) \mathrm{d} s,$$is necessary and sufficient for the Hilbert transform $T$ to be bounded in the weighted space $L^2(W)$.It was well known since early 90 s that $\|T\|_{L^2(W)} \gtrsim[W]_{A_2}^{1 / 2}$ for all weights, and that for some weights $\|T\|_{L^2(W)} \gtrsim[W]_{A_2}$. The famous $A_2$ conjecture (first stated for scalar weights) claims that the second bound is sharp, i.e.$$\|T\|_{L^2(W)} \lesssim[W]_{A_2}$$for all weights.
After some significant developments (and some prizes obtained in the process) the scalar $A_2$ conjecture was finally proved: first by J. Wittwer for Haar multipliers, then by S. Petermichl for Hilbert Transform and for the Riesz transforms, and finally by T. Hytönen for general Calderón-Zygmund operators.
However, while it was a general consensus that the $A_2$ conjecture is true in the matrix case as well, the best known estimate, obtained by Nazarov-Petermichl-Treil-Volberg (for all Calderón-Zygmund operators) was only $\lesssim[W]_{A_2}^{3 / 2}$.
But this upper bound turned out to be sharp! In a recent joint work with K. Domelevo, S. Petermichl and A. Volberg we constructed weights $W$ such that$$\|T\|_{L^2(W)} \gtrsim[W]_{A_2}^{3 / 2},$$so the above exponent $3 / 2$ is a correct one.
In the talk I'll explain motivations, history of the problem, and outline the main ideas of the construction. The construction is quite complicated, but it is an "almost a theorem" that no simple example is possible.
This is joint work with K. Domelevo, S. Petermichl and A. Volberg.[-]

Most classical local properties of a Banach spaces (for example type or cotype, UMD), and most of the more recent questions at the intersection with geometric group theory are defined in terms of the boundedness of vector-valued operators between Lp spaces or their subspaces. It was in fact proved by Hernandez in the early 1980s that this is the case of any property that is stable by Lp direct sums and finite representability. His result can be seen as one direction of a bipolar theorem for a non-linear duality between Banach spaces and operators. I will present the other direction and describe the bipolar of any class of operators for this duality. The talk will be based on my preprint arxiv:2101.07666.[-]