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.[-]

In a recent paper, the speaker and M.I. Ostrovskii developed a new metric embedding method based on the theory of equal-signs-additive (ESA) sequences developed by Brunel and Sucheston in 1970's. This method was used to construct bilipschitz embeddings of diamond and Laakso graphs with an arbitrary finite number of branches into any non-superreflexive Banach space with a uniform bound on distortions that is independent of the number of branches.
In this talk we will outline a proof that the above mentioned embeddability results cannot be obtained using the embedding method which was used for trees by Bourgain (1986) and for binary branching diamonds and Laakso graphs by Johnson and Schechtman (2009), and which is based on a classical James' characterization of superreflexivity (the factorization between the summing basis and the unit vector basis of $\ell_1$). Our proof uses a “self-improvement” argument and the Ramsey theorem.
Joint work with M.I. Ostrovskii.[-]