JOINT COLLOQUIUM Friday 10/12/10 Carslaw 175 2:30pm We will depart for lunch at 1pm from the 2nd floor of Carslaw. Title: "Hölder classification of spheres and Tsar’kov’s phenomenon". Abstract: This talk is in the areas of infinite-dimensional and quantum geometry with some elements of algebra and analysis on infinite-dimensional spaces. The theory of the uniform classification of infinite-dimensional spheres has been developed, mainly, thanks to the solution of the distortion problem by E. Odell and Th. Schlumprecht and is more balanced than the continuous, isometric, Lipschitz or uniform classifications of infinite-dimensional Banach spaces. It allows to transfer a group structure, group actions and other metric-related constructions from one space onto another. In particular, we provide multiple examples of spaces that do not allow any C*-algebra structure but can be endowed with a homogeneous Hölder group structure. We show that the uniformly continuous homeomorphisms can be "upgraded" to the Hölder ones in the classical setting and establish the explicit and, occasionally, sharp exponents of the Hölder regularity for pairs of concrete spaces, including various Besov, Lizorkin-Triebel, Sobolev, sequence, Schatten-von Neumann and other Banach spaces (including lattices and more general non-commutative spaces). These results appear to have close ties with the presence of a remarkable phenomenon from the infinite-dimensional approximation theory discovered by Tsar’kov for the pairs of Lebesgue spaces.