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Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Mathematics of collective behavior and the subject of fluid dynamics undergo a rapid merger where most fascinating breakthroughs have been discovered in recent years. These breakthroughs are based on our novel understanding of collective phenomena from the viewpoint of the laws of fluid motion. Pattern formations in animal swarms or clustering in social networks or even technological applications to decentralized control of unmanned vehicles are just a few examples that can be studied with the new techniques. This workshop will bring together a diversified group of researches from both fields to share the knowledge and promote collaboration in this emerging and fascinating area.

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

Fluids are ubiquitous in the nature, but equations of fluid mechanics are among the most difficult PDEs to analyze. The question of global regularity v.s. finite time blow-up remains open for many fundamental fluid equations, and there are also many other interesting open questions on other aspects of properties. Recently, the study of fluid equations has witnessed multiple breakthrough results, such as non-existence of type-one singularity, rigorous proofs for small scale formation and finite-time blow-up for various equations, non-uniqueness of weak solutions to the Euler and Navier-Stokes equation via convex integration, as well as the non-uniqueness of the law of the 3D stochastic Navier-Stokes equations. With these exciting developments in the last decade, this workshop is a timely event to capitalize on this momentum. In this workshop, we aim to bring together leading experts and junior researchers covering complementary areas in fluid equations to share new ideas, discuss emerging problems, and identify promising research directions for the future. When inviting participants to the workshop, preferences will be given to junior researchers and researchers from underrepresented groups.

Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

The Banff International Research Station will host a diverse, international group of mathematicians in 2023 to study geometric approaches to fluids and optimal transport. The ideas of geodesic settings for fluids have been circulating since the late 1960s, but received a strong surge lately, in the last 5-10 years, with the development of new techniques in infinite-dimensional geometry and group theory. The original idea is that the particles in an ideal liquid or gas move in a curved infinite-dimensional space, in the same way as an airplane flying around the earth: along the shortest path. Properties of the fluid, such as how well it can be predicted in the future, how well it mixes together, and how extreme the speed can be, can all be related to how curved this infinite-dimensional space is, which helps us understand features of the weather and climate geometrically. More recently similar ideas have been used to understand solids as well, for example when a pile of rocks are transported from one place to another at the optimal cost.

Geometry and Topology,    Condensed Matter Physics and Materials