Abstract:Let V be a real normed vector space such that for a fixed 2 <= k < dim V any two k-dimensional subspaces of V are isometric. Is the norm on V necessarily induced by an inner product? This question of Banach is currently known to have an affirmative answer unless k + 1 = dim V = 4m >= 8 or k + 1 = dim V = 134, in which cases the question is open.

After an overview of earlier results I will sketch a proof for the cases k = 2 and k = 3, obtained in a joint work with Sergei Ivanov and Anya Nordskova. In particular, we handle the case k = 3, dim V = 4 which was out of reach of the known global topological methods since the 3-sphere is parallelisable. Our proof is based on a differential-geometric analysis in a neighbourhood of a single k-plane, which also allows us to solve a stronger, local version of the problem.

Bio: Daniil Mamaev is a PhD student at the London School of Geometry and Number Theory under the supervision of Yanki Lekili. His research interests lie in differential geometry and topology, and homological mirror symmetry.

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