Non-unitarisable representations and maximal symmetry

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of ${\rm Aut}(T)$ on $\ell_2(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of ${\rm GL}(\mathcal H)$ extending a well-known non-unitarisable representation of $\mathbb F_\infty$. As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex

Displaying Polish groups on separable Banach spaces

A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology. Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has a display.Comment: 27 page

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces \`a la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size $\aleph_1$ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos

On a Question of Haskell P. Rosenthal Concerning a Characterization of c_0 and ℓ_p

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of c_0 or ℓ_p. An analogous problem concerning spreading models is also addressed

Surface sealing as affected by various rock fragment covers in West Africa

Field studies on the influence of rock fragments on surface sealing, hence infiltration, remain scarce and contradictory. To document this issue, rainfall simulation experiments were carried out on 37 undisturbed 1-m2 plots located along a pedo-climatic transect across West Africa. An important part of the variability of infiltration coefficient (R2 = 0.71) could be explained by a simple model based on the position of rock fragments in the top layer and accounting for the areal percentages of three types of soil surface : (1) bare surface, (2) soil surface covered with rock fragments embedded in the soil surface, and (3) surface with rock fragments resting on top of the soil surface. An even higher determination coefficient (R2 = 0.76) was gained when rock fragment size were accounted for in combination with vesicular porosity. Furthermore, a positive linear relationship was obtained between infiltration coefficient and mean annual rainfall suggesting that additional factors related to climate might be involved, including organic matter content and clay mineralogy. Two main regions could thus be differentiated. In the arid and semi-arid zones, coarse gravel and cobbles embedded in a seal are predominant and generate high runoff. Conversely, fine and medium gravel, mainly free at soil surface, are dominant in the wetter zone, favouring therefore higher infiltration rate. (Résumé d'auteur