Oleg Aispchuk
University of Cincinnati
Title
Rigidity in the Planar Ulam Floating Body Problem for Certain Perimetral Densities
Abstract
In my talk, I will discuss the Ulam Floating Body Problem: “Is a solid of uniform density which will float in water in every position a sphere?”, the Scottish Book. I will focus exclusively on the planar case, which can be formulated by considering the cross-section of a long cylindrical body of uniform relative density $\rho<1$ placed in water.
The planar problem leads to a rich interplay between geometry, differential equations, and the analysis of floating conditions. In particular, I will discuss how the requirement that the body float in equilibrium in every orientation imposes strong geometric constraints on its boundary. These constraints can be expressed in terms of a perimetral density, $\sigma$, which measures the portion of the boundary lying below the waterline in equilibrium.
I will present recent results obtained jointly with Maksim Kosmakov and Pavel Zatitskii for the perimetral densities $\sigma=\frac16,\,\frac18,\,\frac38$.
These results establish new rigidity phenomena for the corresponding planar floating body problem and extend the range of densities for which one can prove that the floating conditions force strong geometric restrictions on the body. I will also briefly discuss the ideas behind the proofs, including the reduction of the geometric problem to a system of differential equations and the role of symmetry and dynamical-systems methods.
