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

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). Collective dynamics is ubiquitous in science and appears in areas like social sciences, biology and neuroscience. Examples encompass the alignment of a flock, the emergence of opinion consensus, or neural synchronization. Most biological systems consist of a large amount of agents (often ranging from a billion to a million units). Hence, uncovering the mechanisms responsible for the cooperation and the spontaneous formation of large-scale self-organized patterns in the full population is a fundamental question with strong scientific implications. It is remarkable that a handful of breakthrough papers appeared in the transition of the millennium, triggering thousands of contributions over the last decade. From a mathematical perspective, the main breakthrough is that unveiling self-organization in a large group of agents can be tackled using strong mathematical methods from nonlinear and nonlocal PDEs. Typically, nonlinear ODEs/PDEs are used to describe the evolution of these complex systems at the various scales of description: microscopic (ODEs), mesoscopic (kinetic PDEs) and macroscopic (fluid-type PDEs). Classical methods have transcended applications and gave rise to strong advances in nonlinear PDE, optimal transport, statistical mechanics or fluid mechanics with seminal works by a large community of international researchers. The main research goal of HICODY was to investigate an emerging class of collective dynamics models describing the evolution of large populations of self-organized agents. More specifically, the cornerstone in HICODY was to develop innovative techniques in the analysis of nonlinear PDEs to go beyond the classical setting with pairwise interactions (PWI) and address more realistic scenarios governed by higher-order interactions (HOI). The common ground in the previous literature was the imposing premise that cooperation is ruled by PWI. However, the up-to-date experimental evidence suggests that PWI are often not sufficient to properly shape collective behavior in real-world systems, and it conjectures that communication rather is a collective nonlinear action at the level of groups. To date, these HOI had remained mostly unexplored due to their mathematical complexity, leaving the experimental community without appropriate models to test hypotheses. The ultimate aim of HICODY was to contribute to filling this gap and providing novel mathematical analysis and computational tools that scientists need to explain complex patterns in nature. This project was divided into three main blocks: 1) The first one aimed to deriving the rigorous kinetic and fluid-type PDEs arising in statistical mechanics from the underlying microscopic description, thus characterizing the evolution of the probability distribution of agents and representing HOI at larger scales. Besides, the formation of patterns from multiple interactions was analyzed in several examples of velocity alignment and synchronization dynamics among others. 2) The second block focused on the study of long-time behavior in a particular integro-differential equation arising in quantitative genetics which describes the dynamics of a population structured by phenotypical traits and governed by Darwinian selection and sexual reproduction. Understanding how patterns emerge from the corresponding recombination of traits during individual interactions is of crucial importance in genetics. 3) Finally, the last block integrated an interdisciplinary approach which combined analytical and computational tools to face the demanding technical level in an innovative application to neuroscience. Specifically, novel activity patterns were explored in large ensembles of neurons with PWI but also HOI.

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