My research interests are in applied analysis and the calculus of variations, a branch of mathematics that studies optimization in infinite-dimensional spaces. A large part of my work focuses on elasticity theory, particularly existence of energy minimizers, pattern formation, scaling laws and rigorous dimension reduction. Currently, with collaborators, I am applying these methods to understand self-assembly, the phenomenon in which small building blocks come together to form large coherent structures.
A second line of research I have been involved in is the study of large graph limits of coupled oscillator networks. My collaborators and I developed a graphon-based framework for studying synchronization in large random networks of coupled oscillators. Graphons are kernels that can be interpreted as continuum limits of adjacency matrices of graphs as their size tends to infinity. This passage to the continuum limit makes the dynamics more analytically tractable while still governing the behaviour of dynamics on large finite networks. In related work we proposed a framework for designing networks robust to perturbations, based on optimizing a spectral measure of network robustness. Using real-world power grids, we studied the problem of optimally placing renewable power generators and allocating susceptance along transmission lines.
I am interested in collaborations where rigorous asymptotics and coarse-graining are used in modelling phenomena in materials science or the analysis of large networked systems.
Please describe one or two of your most interesting projects.
A line of research I am currently focusing on is the analysis of self-assembly. Self-assembly is a ubiquitous process by which living systems spontaneously organize simple building blocks into coherent structures. Inspired by biological organization, it has emerged as a widely used strategy for fabricating materials, with applications in fields such as photonics and medicine. In these applications it is desirable to have a mechanism that limits the size of the self-assembled crystals, and one such mechanism is provided by the curvature of the substrate: whereas crystals grown on flat surfaces can grow without bound, curvature imposes a geometric limit on crystal size. Experiments in which colloidal particles are deposited onto liquid droplets show that, once enough particles are present, the resulting crystals undergo a shape transition from isotropic disc-like forms to highly anisotropic branched structures, even though they assemble slowly enough that energy minimization should apply. Building on the surrounding literature, in joint work with Ian Tobasco we propose a mathematical model for these phenomena and develop a framework for rigorously proving lower bounds on the energy. In the future, we plan to study the evolution of configurations under the gradient flow of this energy and to investigate the role of the energy landscape in selecting the branched configurations observed in nature.
How did you end up where you are today? (Your research journey)
I began my PhD intending to study complex systems and emergent phenomena in large interacting systems. Over the course of it my interests shifted towards analysis and partial differential equations, and my thesis concerned the existence of energy minimizers for models of highly stretchable elastic surfaces. I continued, however, to collaborate on projects involving complex systems, in particular large networks of coupled oscillators, where my background in analysis proved useful in establishing rigorous results about synchronization in large random networks.