Joint Colloquium: Ajiev -- Holder classification of spheres and Tsar’kov’s phenomenon

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.