Configuration Spaces over Non-Smooth Spaces

Configurationssystems of many identical particles in a common environmentdescribe a variety of phenomena in multiple contexts and at any scale: from the interaction of physical bodies to the social behavior of a collective, from grains of sand to galaxies in the universe. Clouds of ions in an electromagnetic field, clusters of stars subject to gravitation, schools of fish, automated vehicles collectively navigating traffic are just some examples. The common understanding behind all such complex systems is that focusing on each of their single constituents does not provide any effective description of the system as a whole: the number of particles in a configuration (be it molecules, individuals, etc.) is so large thatfor all practical purposesit may be considered infinite. The configuration space over a given base spacethe collection of all configurations in a given environmentis the mathematical object apt to address this infinity. In recent years, configuration spaces have been shown to inherit properties of the corresponding base spaces: analytical ones, such as completeness; geometric ones, such as curvature; and stochastic ones, such as the existence of random evolutions of configurations described in terms of the random evolution of their particles subject to a common interaction. State-of-the-art research on the subject is however confined to the case of smooth base spaces, such as the standard three-dimensional Euclidean space or a one-sheeted hyperboloid. The goal of the project is to dramatically expand the scope of the theory to include base spaces accounting for all sorts of singularities, from simple ones, such as cones, to intricated and intriguing ones, such as fractals . This wider class of base spaces is central to real-life applications and includes in particular environments with obstacles (rocks and barrier reefs which shoals of fish stay clear of) and networks (roads and routes which vehicles are confined to). Together with non-smooth base spaces, the project shall also consider rougher interactions described as singular functions of the mutual distance between particles, as well as the random stochastic dynamics of a tagged particle in the system, for example: a predator fish hunting in a shoal, or an electrically charged probe in a plasma.

The research conducted as part of this project has significantly advanced our understanding of the mathematical description of configurations-ensembles of infinitely many identical particles, potentially influenced by physical forces. This abstract framework has broad applicability, ranging from the behavior of gas molecules under electromagnetic interactions to the dynamics of galaxy clusters governed by gravitational forces. Notably, it offers effective constraints on the long-term behavior of such particle systems when random forces arise from energy fluctuations within the system. Mathematically, the project has provided a comprehensive understanding of the concept of "curvature" in both local and non-local dynamics on configuration spaces. The findings demonstrate that these spaces are not flat but instead exhibit lower bounds on an appropriate notion of local curvature, specifically "Ricci curvature." In achieving these results, the project also developed several mathematical tools with extensive applications to other problems, including the study of infinite systems and, more broadly, infinite-dimensional spaces. These newly developed tools include, in particular: representations of the system's evolution as the "sum" of all its dynamically invariant parts. This approach allows a focus on each individual invariant component rather than the system as a whole, thereby reducing the system's complexity to that of its minimal constituents. Additionally, the tools provide a highly general understanding of easily verifiable local conditions for the global convergence of dynamics in the absence of energy fluctuations. They also offer effective synthetic descriptions of curvature lower bounds, formulated in terms of a newly introduced metric for comparing different configurations of the same particle system. Finally, the tools developed during the project have found significant applications in studying the stochastic evolution of density profiles. These tools provide a framework for describing particle systems at the mesoscopic scale-beyond the microscopic level-through stochastic partial differential equations, such as the renowned Dean-Kawasaki equation.