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The Ribe Programme Keith Ball December 6 , 2011 Introduction In 1976

In 1976 Ribe [R] proved that uniformly homeomorphic Banach spaces have uniformly linearly isomorphic finite-dimensional subspaces: that if there is a uniformly continuous homeomorphism between two Banach spaces, with uniformly continuous inverse, then every finite-dimensional subspace of one space, is linearly isomorphic to a subspace of the other, with an implied constant of isomorphism independent of the dimension. This remarkable rigidity theorem guarantees that finite-dimensional properties are determined up to isomorphism by the metric structure of the space. Thus in principle, any property of a space that depends upon finite collections of points has an equivalent formulation which makes no reference to the linear structure of the space and involves only the distances between points. In view of this, Bourgain [Bo1] proposed an ambitious programme which came to be known as “The Ribe Programme”: to find explicit metric descriptions of the most important invariants of normed linear spaces. He himself kick-started the programme by characterising superreflexivity. A Banach space X is called superreflexive if every space whose finite-dimensional spaces embed uniformly well into X, must automatically be reflexive. Bourgain showed this holds precisely if the space does not contain copies of arbitrarily large binary trees. Needless to say the general aim of the Ribe programme is not merely to find metric equivalents of linear properties but to transfer the subtle and well-developed theory of normed spaces to the nonlinear setting, and in this broader sense the programme had been anticipated in a prescient paper of Johnson and Lindenstrauss, [JL]. Within a decade or two the Ribe programme acquired an importance that would have been hard to predict at the outset, as the insights provided by the programme became powerful tools in the theory of algorithms and more recently in geometric