【摘要】
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.
【报告人简介】
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.
Zoom: 890 6551 6489 Password: 447089
