The partial differential equations (PDE) of fluids describe the dynamics of vortices, which appear across a wide range of fluid flows and play an important role in turbulence. Trailing vortices behind aircraft and vortex rings provide familiar examples. One goal of the project is to better understand the behavior of vortices from a mathematical perspective. Another is to study the “typical behavior” of solutions of the basic equations. For example, why are certain scenarios, such as singularity formation, not observed, while we are unable to rule them out mathematically? Are we missing important information about the equations that could be helpful in designing models and in computer simulations of fluid flows? These questions will also be studied in simpler settings that can serve as stepping stones to the full equations. In a somewhat different direction, problems related to PDE of elasticity will also be explored. Together with traditional PDE methods, the project will explore possibilities offered by recent developments in AI for the study of turbulence. The project provides research training opportunities for graduate students. The project focuses on several related directions: vortex dynamics and models for vortex filaments, the formation and evolution of singularities, mechanisms by which singularities may be avoided, and geometric and variational approaches to incompressible flows. Key themes include the emergence of vortex dynamics from the fundamental equations, instabilities and singularity avoidance, the role of random perturbations, and geometric approaches to open problems for the incompressible Euler equations. The project also examines properties of classical energy functionals in variational integrals, including Morrey’s quasiconvexity and its relation to other convexity conditions. Overall, the research aims to identify essential parameters in infinite-dimensional systems. Such an identification can be very helpful for computer simulations of th