CRM: Centro De Giorgi
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Sub-riemannian geometry and beyond, III

On the inverse problem for isometry groups of norms

speaker: Assaf Naor (Princeton University)

abstract: Every Banach space X naturally induces a group G by considering its isometric automorphisms which fix the origin (by the Mazur—Ulam theorem, any such isometry must be linear). The corresponding inverse problem takes as input a group G and asks if it is the isometry group of some equivalent norm on X. Geometrically, when X is finite dimensional this question coincides with asking if there exists an origin-symmetric convex body in Rn whose group of symmetries is G. This talk will present some of the work that has been done on this old question, and then describe recent progress (in collaboration with Breuillard, Liebeck and Rizzoli). It turns out that the answer is positive for some groups but not for others, and it is quite subtle (and still illusive) to obtain an intrinsic characterization of which groups G admit such a representation as the isometry group of a norm. We will describe a new criterion that can be used to obtain a solution of this inverse problem in some settings by using a range of tools from a variety of areas.

Fri 23 Jun, 16:30 - 17:30, Aula Dini
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