Intra-disciplinary bridges for multi-dimensional patterns

The perspective of pattern formation has been successful in drawing from and helping advance multiple areas of mathematics, including dynamical systems, partial differential equations and numerical computing. Formal asymptotic and rigorous approaches such as spatial dynamics have been highly successful over the past years to study/prove the existence and stability of patterns in one spatial dimension. They have also been extended to higher dimensions under certain geometries: such as cylinderical, channel-like domains, etc. They are also useful in understanding invasion fronts, localised patterns, spiral waves and defects in 1D. However, the extension of the wealth of the above mentioned approaches to the analysis of patterns in 2D/3D is not straightforward. 

 

A non-exhaustive list of examples of situations that have proved to be resistant to analysis, and yet very relevant in diverse applications are: patterns formed with more than one preferred lengthscale, aperiodic patterns, multi-dimensional defects, spatial localisation without radial symmetry, patterns in heterogeneous domains, patterns in the presence of a dynamic bifurcation parameter, patterns in lattice systems and non-local systems. However in all of these examples, we are able to obtain numerical approximations to equilibria of the associated governing PDE, either through an initial-boundary value problem approach (time-stepping) or via a root-finding approach (numerical continuation). 

 

Since it is a non-objective function if numerical computability equals proof of existence, I want to explore novel and dimensionally agnostic, intra-disciplinary bridges to pattern formation, that will help us to obtain (using computational algebraic geometry), analyse (using computer assisted proofs as a certification problem) and characterise (using topological data analysis) truly multi-dimensional patterns.