An elegant theorem by J.L. Lions establishes well-posedness of non-autonomous evolutionary problems in Hilbert spaces which are defined by a non-autonomous form. However a regularity problem remained open for many years. We give a survey on positive and negative (partially very recent) results. One of the positive results can be applied to an evolutionary network which has been studied by Dominik Dier and Marjeta Kramar jointly with the speaker. It is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. Besides existence and uniqueness long-time behaviour can be described. When conductivity and diffusion coefficients match (so that mass is conserved) the solutions converge to an equilibrium.[-]

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.[-]

I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then $\mathbb{Z}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space, and one gets from here that there are quite natural nontrivial twisted sums $0\to Z_2\to\mathbb{Z}^{(4)}\to Z_2 \to0$ of $Z_2$ with itself. The twisted sum space $\mathbb{Z}^{(4)}$ does not embeds in, or is a quotient of, a twisted Hilbert space and does not contain $\ell_2$ complemented. We will also construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented. These results have some connection with the nowadays called Kalton calculus [3, 4], and thus several recent advances [2] in this theory that combines twisted sums and interpolation theory will be shown.

Banach space - twisted sum - complex interpolation - Hilbert space[-]