# Colloquium Abstracts

Date/Time/Location: Tuesday Sep 26, 2017 from 3:00-3:50pm in BAC 219

Speaker/Affiliation: Michal Wojciechowski, Institute of Mathematics, Polish Academy of Sciences

Title: On the Auerbach-Mazur-Ulam problem

Abstract: In 1935 Auerbach, Mazur and Ulam proved that any centrally symmetric body in $$\mathbb{R}^3$$ with all two dimensional central sections affinely equivalent to each other is an ellipsoid. This theorem was later generalized to all odd dimensions by Gromov. The proofs are based on the algebraic topology - nonexistence of a non-vanishing vector field tangent to the sphere. This is the reason why in even dimensions the problem is still open - this argument does not work there.

We present a new approach to the problem, that does not use the homological properties of sphere. Under some mild smoothness condition we prove the theorem in 3D using only differential properties of the body. We hope that this approach will work in even dimensions. The talk is based on a joint work with Bartek Zawalski.