Mathematical Challenges of Higher-Order Interactions in Collective Dynamics, and Applications

The mathematical analysis of collective dynamics has experienced a prominent growth in the last years leading to new frontiers with cutting-edge fields in physics, biology and social sciences (e.g. complex networks, active matter or crowd dynamics).

The deep breakthrough is that unveiling self-organization in a large group of agents can be tackled using strong mathematical methods from nonlinear and nonlocal PDEs, like harmonic analysis, energy methods, optimal transport and fluid mechanics. HICODY aims to go beyond the classical restrictive case of pairwise interactions.

Indeed, recent advances in neural networks suggest that higher-order interactions are often needed to properly shape collective dynamics. Classical techniques break down in this setting, thus requiring innovative methods. This proposal is divided into three blocks.

The first one aims to derive the rigorous kinetic and fluid-type PDEs of statistical mechanics from the underlying microscopic description to represent higher-order interactions at larger scales.

Besides, the formation of patterns from multiple interactions will be analyzed in several examples of velocity alignment and synchronization dynamics.

The other blocks integrate an interdisciplinary approach which combines analytical and computational tools to face the demanding technical level in two innovative applications: neuroscience and developmental biology.

First, novel activity patterns will be explored in large ensembles of neurons with multiple interactions; second, a new PDE for filopodia-mediated morphogenesis will be rigorously derived supported by empirical evidence, contrarily to Turings theory based on free diffusion.

As an evidence of the researcher capabilities, he has pioneered techniques to derive hydrodynamic and mean field limits in flocking and synchronization models with pairwise singular interactions. The project will be developed alongside his supervisor, who is expert in nonlinear PDEs and mathematical biology.