My first research project as an undergraduate studied the interplay between reaction-diffusion equations and geometric curvature. Multi-component reaction diffusion equations are known to support Turing pattern - steady-state solutions that are non-uniform in space. Such a mechanism for pattern formation has been proposed in many biological systems. Prior analyses of reaction-diffusion equations generally took place on either flat surfaces or surfaces with uniform curvature, such as a sphere. We analyzed the effects of small deformations away from these simple surfaces.

Standard analysis of reaction-diffusion equations involves performing an expansion around a uniform solution and identifying unstable eigenmodes. The basic mathematical question that we analyzed was: how are the eigenfunctions of the Laplacian (diffusion) operator on a simple surface - a cylinder, a sphere, etc. - modified in the presence of small geometric deformations? We developed a perturbation theory to understand this. These deformed geometries can be obtained from the simpler geometries via a conformal mapping, which modifies the Laplacian operator in a well-defined manner. To leading-order, we find that the effect of the deformation is to introduce an effective potential in the Laplacian of the undeformed surface. This in turn modifies the spectrum of the Laplacian in a straightforward manner, which can be calculated using standard Rayleigh-Schrodinger perturbation theory.

This analysis allows us to rigorously demonstrate the symmetry-breaking properties of geometric deformations. For example, introducing a periodic spatial deformation in a cylinder opens up a band-gap in the Laplacian spectrum. The consequence of this for the resulting Turing patterns is a sharp transition in the phase of the pattern as the unstable eigenmode is tuned past the bandgap, shown below:

This analysis can be performed on a large class of deformations, which we explore in our paper. For example, a localized bump can cause a pinning of the resulting Turing patterns, which we can understand in terms of this conformal perturbation theory. In addition, transient quasi-patterns can be induced upon the inclusion of white noise in the reaction-diffusion equation.