Mathematical solutions to materials problems

This article introduces the Themed Issue “Mathematical Solutions to Materials Problems,” featuring four contributions by eminent professors and leading experts in the field. It outlines how mathematical concepts underpin the description, analysis, and prediction of the behavior of materials across scales. Beginning with basic statistical treatment of experimental data, it traces the role of algebra and tensor calculus in crystalline materials, and topology in disordered and multiphase systems. Variational and phase-field approaches provide a continuum description of evolving microstructures, while discrete graph-based representations capture transport and diffusion processes in atomistic systems. Inverse problems arising in modern microscopy and scattering methods highlight the intimate link between measurement and mathematical reconstruction. Finally, data-driven approaches, including neural operators, are presented as emerging tools connecting large experimental data sets with predictive continuum models. Together, these perspectives demonstrate that mathematics is not merely descriptive, but constitutive of modern materials science.

Neutron tomography of a fossil carbonate from Kentucky illustrates the reconstruction of internal structure via inverse mathematical methods.