Theoretical Statistics and Mathematics Unit, ISI Delhi
Lipschitz geometry involves the study of Banach spaces as a category of objects where, in place of the familiar class of bounded linear maps, the morphisms are taken to be Lipschitz maps and, in some cases, uniformly continuous mappings, arising from the underlying metric/uniform structure attached to the norm of the Banach space. It is rather surprising that in the new setup, there are a number of situations where the linear structure of aBanach space is adequately captured by looking at it as a metric space with the morphisms now consisting of Lipschitz/uniformly continuous maps.A fundamental question pertaining to this circle of ideas and asking whether two given (separable) Banach spaces X and Y are already linearly homeomorphic as soon as they are Lipschitzhomeomorphic, remains open since it was posed for the first time in his pioneering work by Lindenstrauss in the early seventies.
The purpose of this talk is to draw attention to certain issues involving Lipschitz geometry of Banach spaces which lead to new nonlinear characterisations of Hilbert spaces. This will be done mainly in the context of certain issues involving the nonlinear structure of Banach spaces arising from the familiar
(i) The Hahn Banachextension theorem: Existence of a continuous linear ‘extension’ operator involving certain spaces of Lipschitz maps.
(ii) Open mapping theorem: Existence of a Lipschitz lifting (right inverse) for a quotient map.
Time permitting, abrief mention of open problems including directions for further research shall also be made at end of the talk.