Florent Baudier: Nonlinear indices à la Bourgain and applications to coarse and Lipschitz universality

Data: sexta-feira, 25 de Junho de 2021, às 9h30.

Florent P. Baudier (Texas A&M University)

Título: Nonlinear indices à la Bourgain and applications to coarse and Lipschitz universality.

Resumo:  The classical Banach-Mazur theorem states that every separable Banach space admits a linear isometric embedding into C[0,1], the separable Banach space of continuous functions on the unit interval, and we say that C[0,1] is an isometric universal space for the class of separable Banach spaces. In 1968, Szlenk showed that there is no isomorphic separable reflexive Banach space for the class SR of separable reflexive Banach spaces. Szlenk's result was significantly improved by Bourgain in 1980 when he showed that if a Banach space is isomorphic universal for the class SR then it must contain an isomorphic copy of C[0,1]. Bourgain's influential argument relies on a tree ordinal index. In this talk, we introduce nonlinear indices in the spirit of Bourgain's tree indices and show some universality results in the Lipschitz and coarse category. While our Lipschitz universality result is valid in ZFC, one of the coarse universality results requires some additional set-theoretic axioms.  Based on a joint work with G. Lancien (Université Bourgogne Franche-Comté), P. Motakis (York University), and Th. Schlumprecht (Texas A&M University)