The mathematics behind minimalist hats and constraint maps

Professor Henrik Shahgholian will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm.

A classical problem in geometric analysis is the study of constraint maps – mappings between geometric objects that must satisfy given restrictions. A popular example is the problem of using as little fabric as possible to design a fashionable hat. Similar problems arise naturally in contexts where shapes are optimised under constraints. These have been studied in various forms since ancient times, when architects and engineers utilised the principles of minimal surfaces to construct stable and efficient structures, such as arches and domes.

Related ideas can be found in works that were written almost a century ago by Scottish biologist Sir D’Arcy Thompson, who studied the extent to which differences in the shape of closely related animals could be understood as the result of relatively simple mathematical transformations. His ideas about nature’s inherent mathematical structures inspired a many researchers and designers in the natural sciences, art and architecture.

Despite long-standing interest, many fundamental questions remain regarding the mathematical structure of constraint maps. The planned project will study their continuity issues and geometric properties, with a particular focus on how deformations affect the properties of the imaged objects. A key question is whether a deformation necessarily causes discontinuities in the material. For example, mapping a ball that is reshaped as a ring requires that the ball is torn apart, which leads to interruptions or singularities. One of the goals of the project is to identify conditions that exclude the occurrence of such singularities and, when they do occur, to describe their structure and the behaviour of the mapping in their vicinity.