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## Any isometry between the spheres of 2-dimensional Banach spaces is linear

Abstract: The famous Tingley’s Problem asks whether every onto isometry between the unit spheres of Banach spaces can be extended to an isometry between the spaces – in this case, the Mazur–Ulam Theorem ensures that the isometry between the spaces would be linear. This Problem is far from been solved, but at least the two-dimensional case has been solved recently in a series of works by the author and Professor Tarás Banakh. In this seminar we will show how a miscellany of several kinds of tools (single variable calculus, linear algebra, differential equations) has allowed to prove the following statement: If X and are two-dimensional Banach spaces, then every isometry between their unit spheres extends to an isometry between X and Y.